*> \brief \b DGEBAK
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEBAK + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEBAK( JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOB, SIDE
* INTEGER IHI, ILO, INFO, LDV, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION SCALE( * ), V( LDV, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEBAK forms the right or left eigenvectors of a real general matrix
*> by backward transformation on the computed eigenvectors of the
*> balanced matrix output by DGEBAL.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> Specifies the type of backward transformation required:
*> = 'N', do nothing, return immediately;
*> = 'P', do backward transformation for permutation only;
*> = 'S', do backward transformation for scaling only;
*> = 'B', do backward transformations for both permutation and
*> scaling.
*> JOB must be the same as the argument JOB supplied to DGEBAL.
*> \endverbatim
*>
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'R': V contains right eigenvectors;
*> = 'L': V contains left eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows of the matrix V. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> The integers ILO and IHI determined by DGEBAL.
*> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*> \endverbatim
*>
*> \param[in] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION array, dimension (N)
*> Details of the permutation and scaling factors, as returned
*> by DGEBAL.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of columns of the matrix V. M >= 0.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,M)
*> On entry, the matrix of right or left eigenvectors to be
*> transformed, as returned by DHSEIN or DTREVC.
*> On exit, V is overwritten by the transformed eigenvectors.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V. LDV >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleGEcomputational
*
* =====================================================================
SUBROUTINE DGEBAK( JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV,
$ INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER JOB, SIDE
INTEGER IHI, ILO, INFO, LDV, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION SCALE( * ), V( LDV, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LEFTV, RIGHTV
INTEGER I, II, K
DOUBLE PRECISION S
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Decode and Test the input parameters
*
RIGHTV = LSAME( SIDE, 'R' )
LEFTV = LSAME( SIDE, 'L' )
*
INFO = 0
IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND.
$ .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN
INFO = -1
ELSE IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -5
ELSE IF( M.LT.0 ) THEN
INFO = -7
ELSE IF( LDV.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEBAK', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
IF( M.EQ.0 )
$ RETURN
IF( LSAME( JOB, 'N' ) )
$ RETURN
*
IF( ILO.EQ.IHI )
$ GO TO 30
*
* Backward balance
*
IF( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN
*
IF( RIGHTV ) THEN
DO 10 I = ILO, IHI
S = SCALE( I )
CALL DSCAL( M, S, V( I, 1 ), LDV )
10 CONTINUE
END IF
*
IF( LEFTV ) THEN
DO 20 I = ILO, IHI
S = ONE / SCALE( I )
CALL DSCAL( M, S, V( I, 1 ), LDV )
20 CONTINUE
END IF
*
END IF
*
* Backward permutation
*
* For I = ILO-1 step -1 until 1,
* IHI+1 step 1 until N do --
*
30 CONTINUE
IF( LSAME( JOB, 'P' ) .OR. LSAME( JOB, 'B' ) ) THEN
IF( RIGHTV ) THEN
DO 40 II = 1, N
I = II
IF( I.GE.ILO .AND. I.LE.IHI )
$ GO TO 40
IF( I.LT.ILO )
$ I = ILO - II
K = SCALE( I )
IF( K.EQ.I )
$ GO TO 40
CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
40 CONTINUE
END IF
*
IF( LEFTV ) THEN
DO 50 II = 1, N
I = II
IF( I.GE.ILO .AND. I.LE.IHI )
$ GO TO 50
IF( I.LT.ILO )
$ I = ILO - II
K = SCALE( I )
IF( K.EQ.I )
$ GO TO 50
CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
50 CONTINUE
END IF
END IF
*
RETURN
*
* End of DGEBAK
*
END
*> \brief \b DGEBAL
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEBAL + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOB
* INTEGER IHI, ILO, INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), SCALE( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEBAL balances a general real matrix A. This involves, first,
*> permuting A by a similarity transformation to isolate eigenvalues
*> in the first 1 to ILO-1 and last IHI+1 to N elements on the
*> diagonal; and second, applying a diagonal similarity transformation
*> to rows and columns ILO to IHI to make the rows and columns as
*> close in norm as possible. Both steps are optional.
*>
*> Balancing may reduce the 1-norm of the matrix, and improve the
*> accuracy of the computed eigenvalues and/or eigenvectors.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> Specifies the operations to be performed on A:
*> = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
*> for i = 1,...,N;
*> = 'P': permute only;
*> = 'S': scale only;
*> = 'B': both permute and scale.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE array, dimension (LDA,N)
*> On entry, the input matrix A.
*> On exit, A is overwritten by the balanced matrix.
*> If JOB = 'N', A is not referenced.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*> \param[out] IHI
*> \verbatim
*> IHI is INTEGER
*> ILO and IHI are set to integers such that on exit
*> A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
*> If JOB = 'N' or 'S', ILO = 1 and IHI = N.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE array, dimension (N)
*> Details of the permutations and scaling factors applied to
*> A. If P(j) is the index of the row and column interchanged
*> with row and column j and D(j) is the scaling factor
*> applied to row and column j, then
*> SCALE(j) = P(j) for j = 1,...,ILO-1
*> = D(j) for j = ILO,...,IHI
*> = P(j) for j = IHI+1,...,N.
*> The order in which the interchanges are made is N to IHI+1,
*> then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2015
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The permutations consist of row and column interchanges which put
*> the matrix in the form
*>
*> ( T1 X Y )
*> P A P = ( 0 B Z )
*> ( 0 0 T2 )
*>
*> where T1 and T2 are upper triangular matrices whose eigenvalues lie
*> along the diagonal. The column indices ILO and IHI mark the starting
*> and ending columns of the submatrix B. Balancing consists of applying
*> a diagonal similarity transformation inv(D) * B * D to make the
*> 1-norms of each row of B and its corresponding column nearly equal.
*> The output matrix is
*>
*> ( T1 X*D Y )
*> ( 0 inv(D)*B*D inv(D)*Z ).
*> ( 0 0 T2 )
*>
*> Information about the permutations P and the diagonal matrix D is
*> returned in the vector SCALE.
*>
*> This subroutine is based on the EISPACK routine BALANC.
*>
*> Modified by Tzu-Yi Chen, Computer Science Division, University of
*> California at Berkeley, USA
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
*
* -- LAPACK computational routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2015
*
* .. Scalar Arguments ..
CHARACTER JOB
INTEGER IHI, ILO, INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), SCALE( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION SCLFAC
PARAMETER ( SCLFAC = 2.0D+0 )
DOUBLE PRECISION FACTOR
PARAMETER ( FACTOR = 0.95D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOCONV
INTEGER I, ICA, IEXC, IRA, J, K, L, M
DOUBLE PRECISION C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1,
$ SFMIN2
* ..
* .. External Functions ..
LOGICAL DISNAN, LSAME
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DNRM2
EXTERNAL DISNAN, LSAME, IDAMAX, DLAMCH, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* Test the input parameters
*
INFO = 0
IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND.
$ .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEBAL', -INFO )
RETURN
END IF
*
K = 1
L = N
*
IF( N.EQ.0 )
$ GO TO 210
*
IF( LSAME( JOB, 'N' ) ) THEN
DO 10 I = 1, N
SCALE( I ) = ONE
10 CONTINUE
GO TO 210
END IF
*
IF( LSAME( JOB, 'S' ) )
$ GO TO 120
*
* Permutation to isolate eigenvalues if possible
*
GO TO 50
*
* Row and column exchange.
*
20 CONTINUE
SCALE( M ) = J
IF( J.EQ.M )
$ GO TO 30
*
CALL DSWAP( L, A( 1, J ), 1, A( 1, M ), 1 )
CALL DSWAP( N-K+1, A( J, K ), LDA, A( M, K ), LDA )
*
30 CONTINUE
GO TO ( 40, 80 )IEXC
*
* Search for rows isolating an eigenvalue and push them down.
*
40 CONTINUE
IF( L.EQ.1 )
$ GO TO 210
L = L - 1
*
50 CONTINUE
DO 70 J = L, 1, -1
*
DO 60 I = 1, L
IF( I.EQ.J )
$ GO TO 60
IF( A( J, I ).NE.ZERO )
$ GO TO 70
60 CONTINUE
*
M = L
IEXC = 1
GO TO 20
70 CONTINUE
*
GO TO 90
*
* Search for columns isolating an eigenvalue and push them left.
*
80 CONTINUE
K = K + 1
*
90 CONTINUE
DO 110 J = K, L
*
DO 100 I = K, L
IF( I.EQ.J )
$ GO TO 100
IF( A( I, J ).NE.ZERO )
$ GO TO 110
100 CONTINUE
*
M = K
IEXC = 2
GO TO 20
110 CONTINUE
*
120 CONTINUE
DO 130 I = K, L
SCALE( I ) = ONE
130 CONTINUE
*
IF( LSAME( JOB, 'P' ) )
$ GO TO 210
*
* Balance the submatrix in rows K to L.
*
* Iterative loop for norm reduction
*
SFMIN1 = DLAMCH( 'S' ) / DLAMCH( 'P' )
SFMAX1 = ONE / SFMIN1
SFMIN2 = SFMIN1*SCLFAC
SFMAX2 = ONE / SFMIN2
*
140 CONTINUE
NOCONV = .FALSE.
*
DO 200 I = K, L
*
C = DNRM2( L-K+1, A( K, I ), 1 )
R = DNRM2( L-K+1, A( I, K ), LDA )
ICA = IDAMAX( L, A( 1, I ), 1 )
CA = ABS( A( ICA, I ) )
IRA = IDAMAX( N-K+1, A( I, K ), LDA )
RA = ABS( A( I, IRA+K-1 ) )
*
* Guard against zero C or R due to underflow.
*
IF( C.EQ.ZERO .OR. R.EQ.ZERO )
$ GO TO 200
G = R / SCLFAC
F = ONE
S = C + R
160 CONTINUE
IF( C.GE.G .OR. MAX( F, C, CA ).GE.SFMAX2 .OR.
$ MIN( R, G, RA ).LE.SFMIN2 )GO TO 170
IF( DISNAN( C+F+CA+R+G+RA ) ) THEN
*
* Exit if NaN to avoid infinite loop
*
INFO = -3
CALL XERBLA( 'DGEBAL', -INFO )
RETURN
END IF
F = F*SCLFAC
C = C*SCLFAC
CA = CA*SCLFAC
R = R / SCLFAC
G = G / SCLFAC
RA = RA / SCLFAC
GO TO 160
*
170 CONTINUE
G = C / SCLFAC
180 CONTINUE
IF( G.LT.R .OR. MAX( R, RA ).GE.SFMAX2 .OR.
$ MIN( F, C, G, CA ).LE.SFMIN2 )GO TO 190
F = F / SCLFAC
C = C / SCLFAC
G = G / SCLFAC
CA = CA / SCLFAC
R = R*SCLFAC
RA = RA*SCLFAC
GO TO 180
*
* Now balance.
*
190 CONTINUE
IF( ( C+R ).GE.FACTOR*S )
$ GO TO 200
IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN
IF( F*SCALE( I ).LE.SFMIN1 )
$ GO TO 200
END IF
IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN
IF( SCALE( I ).GE.SFMAX1 / F )
$ GO TO 200
END IF
G = ONE / F
SCALE( I ) = SCALE( I )*F
NOCONV = .TRUE.
*
CALL DSCAL( N-K+1, G, A( I, K ), LDA )
CALL DSCAL( L, F, A( 1, I ), 1 )
*
200 CONTINUE
*
IF( NOCONV )
$ GO TO 140
*
210 CONTINUE
ILO = K
IHI = L
*
RETURN
*
* End of DGEBAL
*
END
*> \brief DGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEEV + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,
* LDVR, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVL, JOBVR
* INTEGER INFO, LDA, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
* $ WI( * ), WORK( * ), WR( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEEV computes for an N-by-N real nonsymmetric matrix A, the
*> eigenvalues and, optionally, the left and/or right eigenvectors.
*>
*> The right eigenvector v(j) of A satisfies
*> A * v(j) = lambda(j) * v(j)
*> where lambda(j) is its eigenvalue.
*> The left eigenvector u(j) of A satisfies
*> u(j)**H * A = lambda(j) * u(j)**H
*> where u(j)**H denotes the conjugate-transpose of u(j).
*>
*> The computed eigenvectors are normalized to have Euclidean norm
*> equal to 1 and largest component real.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVL
*> \verbatim
*> JOBVL is CHARACTER*1
*> = 'N': left eigenvectors of A are not computed;
*> = 'V': left eigenvectors of A are computed.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*> JOBVR is CHARACTER*1
*> = 'N': right eigenvectors of A are not computed;
*> = 'V': right eigenvectors of A are computed.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the N-by-N matrix A.
*> On exit, A has been overwritten.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (N)
*> WR and WI contain the real and imaginary parts,
*> respectively, of the computed eigenvalues. Complex
*> conjugate pairs of eigenvalues appear consecutively
*> with the eigenvalue having the positive imaginary part
*> first.
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,N)
*> If JOBVL = 'V', the left eigenvectors u(j) are stored one
*> after another in the columns of VL, in the same order
*> as their eigenvalues.
*> If JOBVL = 'N', VL is not referenced.
*> If the j-th eigenvalue is real, then u(j) = VL(:,j),
*> the j-th column of VL.
*> If the j-th and (j+1)-st eigenvalues form a complex
*> conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
*> u(j+1) = VL(:,j) - i*VL(:,j+1).
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the array VL. LDVL >= 1; if
*> JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,N)
*> If JOBVR = 'V', the right eigenvectors v(j) are stored one
*> after another in the columns of VR, in the same order
*> as their eigenvalues.
*> If JOBVR = 'N', VR is not referenced.
*> If the j-th eigenvalue is real, then v(j) = VR(:,j),
*> the j-th column of VR.
*> If the j-th and (j+1)-st eigenvalues form a complex
*> conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
*> v(j+1) = VR(:,j) - i*VR(:,j+1).
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the array VR. LDVR >= 1; if
*> JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,3*N), and
*> if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N. For good
*> performance, LWORK must generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = i, the QR algorithm failed to compute all the
*> eigenvalues, and no eigenvectors have been computed;
*> elements i+1:N of WR and WI contain eigenvalues which
*> have converged.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
* @precisions fortran d -> s
*
*> \ingroup doubleGEeigen
*
* =====================================================================
SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,
$ LDVR, WORK, LWORK, INFO )
implicit none
*
* -- LAPACK driver routine (version 3.6.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
$ WI( * ), WORK( * ), WR( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, SCALEA, WANTVL, WANTVR
CHARACTER SIDE
INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, ITAU, IWRK, K,
$ LWORK_TREVC, MAXWRK, MINWRK, NOUT
DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
$ SN
* ..
* .. Local Arrays ..
LOGICAL SELECT( 1 )
DOUBLE PRECISION DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLABAD, DLACPY,
$ DLARTG, DLASCL, DORGHR, DROT, DSCAL, DTREVC3,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX, ILAENV
DOUBLE PRECISION DLAMCH, DLANGE, DLAPY2, DNRM2
EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE, DLAPY2,
$ DNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
WANTVL = LSAME( JOBVL, 'V' )
WANTVR = LSAME( JOBVR, 'V' )
IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
INFO = -1
ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
INFO = -9
ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
INFO = -11
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV.
* HSWORK refers to the workspace preferred by DHSEQR, as
* calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
* the worst case.)
*
IF( INFO.EQ.0 ) THEN
IF( N.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
ELSE
MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
IF( WANTVL ) THEN
MINWRK = 4*N
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
$ 'DORGHR', ' ', N, 1, N, -1 ) )
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL,
$ WORK, -1, INFO )
HSWORK = INT( WORK(1) )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
CALL DTREVC3( 'L', 'B', SELECT, N, A, LDA,
$ VL, LDVL, VR, LDVR, N, NOUT,
$ WORK, -1, IERR )
LWORK_TREVC = INT( WORK(1) )
MAXWRK = MAX( MAXWRK, N + LWORK_TREVC )
MAXWRK = MAX( MAXWRK, 4*N )
ELSE IF( WANTVR ) THEN
MINWRK = 4*N
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
$ 'DORGHR', ' ', N, 1, N, -1 ) )
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR,
$ WORK, -1, INFO )
HSWORK = INT( WORK(1) )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
CALL DTREVC3( 'R', 'B', SELECT, N, A, LDA,
$ VL, LDVL, VR, LDVR, N, NOUT,
$ WORK, -1, IERR )
LWORK_TREVC = INT( WORK(1) )
MAXWRK = MAX( MAXWRK, N + LWORK_TREVC )
MAXWRK = MAX( MAXWRK, 4*N )
ELSE
MINWRK = 3*N
CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR, LDVR,
$ WORK, -1, INFO )
HSWORK = INT( WORK(1) )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEEV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
CALL DLABAD( SMLNUM, BIGNUM )
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
SCALEA = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
SCALEA = .TRUE.
CSCALE = SMLNUM
ELSE IF( ANRM.GT.BIGNUM ) THEN
SCALEA = .TRUE.
CSCALE = BIGNUM
END IF
IF( SCALEA )
$ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
* Balance the matrix
* (Workspace: need N)
*
IBAL = 1
CALL DGEBAL( 'B', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )
*
* Reduce to upper Hessenberg form
* (Workspace: need 3*N, prefer 2*N+N*NB)
*
ITAU = IBAL + N
IWRK = ITAU + N
CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
IF( WANTVL ) THEN
*
* Want left eigenvectors
* Copy Householder vectors to VL
*
SIDE = 'L'
CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL )
*
* Generate orthogonal matrix in VL
* (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
* Perform QR iteration, accumulating Schur vectors in VL
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
*
IF( WANTVR ) THEN
*
* Want left and right eigenvectors
* Copy Schur vectors to VR
*
SIDE = 'B'
CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
END IF
*
ELSE IF( WANTVR ) THEN
*
* Want right eigenvectors
* Copy Householder vectors to VR
*
SIDE = 'R'
CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR )
*
* Generate orthogonal matrix in VR
* (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
* Perform QR iteration, accumulating Schur vectors in VR
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
*
ELSE
*
* Compute eigenvalues only
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
END IF
*
* If INFO .NE. 0 from DHSEQR, then quit
*
IF( INFO.NE.0 )
$ GO TO 50
*
IF( WANTVL .OR. WANTVR ) THEN
*
* Compute left and/or right eigenvectors
* (Workspace: need 4*N, prefer N + N + 2*N*NB)
*
CALL DTREVC3( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
$ N, NOUT, WORK( IWRK ), LWORK-IWRK+1, IERR )
END IF
*
IF( WANTVL ) THEN
*
* Undo balancing of left eigenvectors
* (Workspace: need N)
*
CALL DGEBAK( 'B', 'L', N, ILO, IHI, WORK( IBAL ), N, VL, LDVL,
$ IERR )
*
* Normalize left eigenvectors and make largest component real
*
DO 20 I = 1, N
IF( WI( I ).EQ.ZERO ) THEN
SCL = ONE / DNRM2( N, VL( 1, I ), 1 )
CALL DSCAL( N, SCL, VL( 1, I ), 1 )
ELSE IF( WI( I ).GT.ZERO ) THEN
SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ),
$ DNRM2( N, VL( 1, I+1 ), 1 ) )
CALL DSCAL( N, SCL, VL( 1, I ), 1 )
CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 )
DO 10 K = 1, N
WORK( IWRK+K-1 ) = VL( K, I )**2 + VL( K, I+1 )**2
10 CONTINUE
K = IDAMAX( N, WORK( IWRK ), 1 )
CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN )
VL( K, I+1 ) = ZERO
END IF
20 CONTINUE
END IF
*
IF( WANTVR ) THEN
*
* Undo balancing of right eigenvectors
* (Workspace: need N)
*
CALL DGEBAK( 'B', 'R', N, ILO, IHI, WORK( IBAL ), N, VR, LDVR,
$ IERR )
*
* Normalize right eigenvectors and make largest component real
*
DO 40 I = 1, N
IF( WI( I ).EQ.ZERO ) THEN
SCL = ONE / DNRM2( N, VR( 1, I ), 1 )
CALL DSCAL( N, SCL, VR( 1, I ), 1 )
ELSE IF( WI( I ).GT.ZERO ) THEN
SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ),
$ DNRM2( N, VR( 1, I+1 ), 1 ) )
CALL DSCAL( N, SCL, VR( 1, I ), 1 )
CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 )
DO 30 K = 1, N
WORK( IWRK+K-1 ) = VR( K, I )**2 + VR( K, I+1 )**2
30 CONTINUE
K = IDAMAX( N, WORK( IWRK ), 1 )
CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN )
VR( K, I+1 ) = ZERO
END IF
40 CONTINUE
END IF
*
* Undo scaling if necessary
*
50 CONTINUE
IF( SCALEA ) THEN
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ),
$ MAX( N-INFO, 1 ), IERR )
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ),
$ MAX( N-INFO, 1 ), IERR )
IF( INFO.GT.0 ) THEN
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
$ IERR )
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
$ IERR )
END IF
END IF
*
WORK( 1 ) = MAXWRK
RETURN
*
* End of DGEEV
*
END
*> \brief \b DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEHD2 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, ILO, INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
*> an orthogonal similarity transformation: Q**T * A * Q = H .
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*>
*> It is assumed that A is already upper triangular in rows
*> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
*> set by a previous call to DGEBAL; otherwise they should be
*> set to 1 and N respectively. See Further Details.
*> 1 <= ILO <= IHI <= max(1,N).
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the n by n general matrix to be reduced.
*> On exit, the upper triangle and the first subdiagonal of A
*> are overwritten with the upper Hessenberg matrix H, and the
*> elements below the first subdiagonal, with the array TAU,
*> represent the orthogonal matrix Q as a product of elementary
*> reflectors. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of (ihi-ilo) elementary
*> reflectors
*>
*> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
*> exit in A(i+2:ihi,i), and tau in TAU(i).
*>
*> The contents of A are illustrated by the following example, with
*> n = 7, ilo = 2 and ihi = 6:
*>
*> on entry, on exit,
*>
*> ( a a a a a a a ) ( a a h h h h a )
*> ( a a a a a a ) ( a h h h h a )
*> ( a a a a a a ) ( h h h h h h )
*> ( a a a a a a ) ( v2 h h h h h )
*> ( a a a a a a ) ( v2 v3 h h h h )
*> ( a a a a a a ) ( v2 v3 v4 h h h )
*> ( a ) ( a )
*>
*> where a denotes an element of the original matrix A, h denotes a
*> modified element of the upper Hessenberg matrix H, and vi denotes an
*> element of the vector defining H(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION AII
* ..
* .. External Subroutines ..
EXTERNAL DLARF, DLARFG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -2
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEHD2', -INFO )
RETURN
END IF
*
DO 10 I = ILO, IHI - 1
*
* Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
*
CALL DLARFG( IHI-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
$ TAU( I ) )
AII = A( I+1, I )
A( I+1, I ) = ONE
*
* Apply H(i) to A(1:ihi,i+1:ihi) from the right
*
CALL DLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ),
$ A( 1, I+1 ), LDA, WORK )
*
* Apply H(i) to A(i+1:ihi,i+1:n) from the left
*
CALL DLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1, TAU( I ),
$ A( I+1, I+1 ), LDA, WORK )
*
A( I+1, I ) = AII
10 CONTINUE
*
RETURN
*
* End of DGEHD2
*
END
*> \brief \b DGEHRD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEHRD + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, ILO, INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEHRD reduces a real general matrix A to upper Hessenberg form H by
*> an orthogonal similarity transformation: Q**T * A * Q = H .
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*>
*> It is assumed that A is already upper triangular in rows
*> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
*> set by a previous call to DGEBAL; otherwise they should be
*> set to 1 and N respectively. See Further Details.
*> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the N-by-N general matrix to be reduced.
*> On exit, the upper triangle and the first subdiagonal of A
*> are overwritten with the upper Hessenberg matrix H, and the
*> elements below the first subdiagonal, with the array TAU,
*> represent the orthogonal matrix Q as a product of elementary
*> reflectors. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> The scalar factors of the elementary reflectors (see Further
*> Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
*> zero.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of the array WORK. LWORK >= max(1,N).
*> For good performance, LWORK should generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2015
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of (ihi-ilo) elementary
*> reflectors
*>
*> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
*> exit in A(i+2:ihi,i), and tau in TAU(i).
*>
*> The contents of A are illustrated by the following example, with
*> n = 7, ilo = 2 and ihi = 6:
*>
*> on entry, on exit,
*>
*> ( a a a a a a a ) ( a a h h h h a )
*> ( a a a a a a ) ( a h h h h a )
*> ( a a a a a a ) ( h h h h h h )
*> ( a a a a a a ) ( v2 h h h h h )
*> ( a a a a a a ) ( v2 v3 h h h h )
*> ( a a a a a a ) ( v2 v3 v4 h h h )
*> ( a ) ( a )
*>
*> where a denotes an element of the original matrix A, h denotes a
*> modified element of the upper Hessenberg matrix H, and vi denotes an
*> element of the vector defining H(i).
*>
*> This file is a slight modification of LAPACK-3.0's DGEHRD
*> subroutine incorporating improvements proposed by Quintana-Orti and
*> Van de Geijn (2006). (See DLAHR2.)
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2015
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER NBMAX, LDT, TSIZE
PARAMETER ( NBMAX = 64, LDT = NBMAX+1,
$ TSIZE = LDT*NBMAX )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0,
$ ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWT, J, LDWORK, LWKOPT, NB,
$ NBMIN, NH, NX
DOUBLE PRECISION EI
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DGEHD2, DGEMM, DLAHR2, DLARFB, DTRMM,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -2
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Compute the workspace requirements
*
NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
LWKOPT = N*NB + TSIZE
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEHRD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Set elements 1:ILO-1 and IHI:N-1 of TAU to zero
*
DO 10 I = 1, ILO - 1
TAU( I ) = ZERO
10 CONTINUE
DO 20 I = MAX( 1, IHI ), N - 1
TAU( I ) = ZERO
20 CONTINUE
*
* Quick return if possible
*
NH = IHI - ILO + 1
IF( NH.LE.1 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
* Determine the block size
*
NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
NBMIN = 2
IF( NB.GT.1 .AND. NB.LT.NH ) THEN
*
* Determine when to cross over from blocked to unblocked code
* (last block is always handled by unblocked code)
*
NX = MAX( NB, ILAENV( 3, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
IF( NX.LT.NH ) THEN
*
* Determine if workspace is large enough for blocked code
*
IF( LWORK.LT.N*NB+TSIZE ) THEN
*
* Not enough workspace to use optimal NB: determine the
* minimum value of NB, and reduce NB or force use of
* unblocked code
*
NBMIN = MAX( 2, ILAENV( 2, 'DGEHRD', ' ', N, ILO, IHI,
$ -1 ) )
IF( LWORK.GE.(N*NBMIN + TSIZE) ) THEN
NB = (LWORK-TSIZE) / N
ELSE
NB = 1
END IF
END IF
END IF
END IF
LDWORK = N
*
IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN
*
* Use unblocked code below
*
I = ILO
*
ELSE
*
* Use blocked code
*
IWT = 1 + N*NB
DO 40 I = ILO, IHI - 1 - NX, NB
IB = MIN( NB, IHI-I )
*
* Reduce columns i:i+ib-1 to Hessenberg form, returning the
* matrices V and T of the block reflector H = I - V*T*V**T
* which performs the reduction, and also the matrix Y = A*V*T
*
CALL DLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ),
$ WORK( IWT ), LDT, WORK, LDWORK )
*
* Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
* right, computing A := A - Y * V**T. V(i+ib,ib-1) must be set
* to 1
*
EI = A( I+IB, I+IB-1 )
A( I+IB, I+IB-1 ) = ONE
CALL DGEMM( 'No transpose', 'Transpose',
$ IHI, IHI-I-IB+1,
$ IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE,
$ A( 1, I+IB ), LDA )
A( I+IB, I+IB-1 ) = EI
*
* Apply the block reflector H to A(1:i,i+1:i+ib-1) from the
* right
*
CALL DTRMM( 'Right', 'Lower', 'Transpose',
$ 'Unit', I, IB-1,
$ ONE, A( I+1, I ), LDA, WORK, LDWORK )
DO 30 J = 0, IB-2
CALL DAXPY( I, -ONE, WORK( LDWORK*J+1 ), 1,
$ A( 1, I+J+1 ), 1 )
30 CONTINUE
*
* Apply the block reflector H to A(i+1:ihi,i+ib:n) from the
* left
*
CALL DLARFB( 'Left', 'Transpose', 'Forward',
$ 'Columnwise',
$ IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA,
$ WORK( IWT ), LDT, A( I+1, I+IB ), LDA,
$ WORK, LDWORK )
40 CONTINUE
END IF
*
* Use unblocked code to reduce the rest of the matrix
*
CALL DGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO )
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DGEHRD
*
END
*> \brief \b DHSEQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DHSEQR + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
* LDZ, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
* CHARACTER COMPZ, JOB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DHSEQR computes the eigenvalues of a Hessenberg matrix H
*> and, optionally, the matrices T and Z from the Schur decomposition
*> H = Z T Z**T, where T is an upper quasi-triangular matrix (the
*> Schur form), and Z is the orthogonal matrix of Schur vectors.
*>
*> Optionally Z may be postmultiplied into an input orthogonal
*> matrix Q so that this routine can give the Schur factorization
*> of a matrix A which has been reduced to the Hessenberg form H
*> by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> = 'E': compute eigenvalues only;
*> = 'S': compute eigenvalues and the Schur form T.
*> \endverbatim
*>
*> \param[in] COMPZ
*> \verbatim
*> COMPZ is CHARACTER*1
*> = 'N': no Schur vectors are computed;
*> = 'I': Z is initialized to the unit matrix and the matrix Z
*> of Schur vectors of H is returned;
*> = 'V': Z must contain an orthogonal matrix Q on entry, and
*> the product Q*Z is returned.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N .GE. 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*>
*> It is assumed that H is already upper triangular in rows
*> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
*> set by a previous call to DGEBAL, and then passed to ZGEHRD
*> when the matrix output by DGEBAL is reduced to Hessenberg
*> form. Otherwise ILO and IHI should be set to 1 and N
*> respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
*> If N = 0, then ILO = 1 and IHI = 0.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On entry, the upper Hessenberg matrix H.
*> On exit, if INFO = 0 and JOB = 'S', then H contains the
*> upper quasi-triangular matrix T from the Schur decomposition
*> (the Schur form); 2-by-2 diagonal blocks (corresponding to
*> complex conjugate pairs of eigenvalues) are returned in
*> standard form, with H(i,i) = H(i+1,i+1) and
*> H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the
*> contents of H are unspecified on exit. (The output value of
*> H when INFO.GT.0 is given under the description of INFO
*> below.)
*>
*> Unlike earlier versions of DHSEQR, this subroutine may
*> explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
*> or j = IHI+1, IHI+2, ... N.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH .GE. max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (N)
*>
*> The real and imaginary parts, respectively, of the computed
*> eigenvalues. If two eigenvalues are computed as a complex
*> conjugate pair, they are stored in consecutive elements of
*> WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and
*> WI(i+1) .LT. 0. If JOB = 'S', the eigenvalues are stored in
*> the same order as on the diagonal of the Schur form returned
*> in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2
*> diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
*> WI(i+1) = -WI(i).
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> If COMPZ = 'N', Z is not referenced.
*> If COMPZ = 'I', on entry Z need not be set and on exit,
*> if INFO = 0, Z contains the orthogonal matrix Z of the Schur
*> vectors of H. If COMPZ = 'V', on entry Z must contain an
*> N-by-N matrix Q, which is assumed to be equal to the unit
*> matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
*> if INFO = 0, Z contains Q*Z.
*> Normally Q is the orthogonal matrix generated by DORGHR
*> after the call to DGEHRD which formed the Hessenberg matrix
*> H. (The output value of Z when INFO.GT.0 is given under
*> the description of INFO below.)
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. if COMPZ = 'I' or
*> COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> On exit, if INFO = 0, WORK(1) returns an estimate of
*> the optimal value for LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK .GE. max(1,N)
*> is sufficient and delivers very good and sometimes
*> optimal performance. However, LWORK as large as 11*N
*> may be required for optimal performance. A workspace
*> query is recommended to determine the optimal workspace
*> size.
*>
*> If LWORK = -1, then DHSEQR does a workspace query.
*> In this case, DHSEQR checks the input parameters and
*> estimates the optimal workspace size for the given
*> values of N, ILO and IHI. The estimate is returned
*> in WORK(1). No error message related to LWORK is
*> issued by XERBLA. Neither H nor Z are accessed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> .LT. 0: if INFO = -i, the i-th argument had an illegal
*> value
*> .GT. 0: if INFO = i, DHSEQR failed to compute all of
*> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
*> and WI contain those eigenvalues which have been
*> successfully computed. (Failures are rare.)
*>
*> If INFO .GT. 0 and JOB = 'E', then on exit, the
*> remaining unconverged eigenvalues are the eigen-
*> values of the upper Hessenberg matrix rows and
*> columns ILO through INFO of the final, output
*> value of H.
*>
*> If INFO .GT. 0 and JOB = 'S', then on exit
*>
*> (*) (initial value of H)*U = U*(final value of H)
*>
*> where U is an orthogonal matrix. The final
*> value of H is upper Hessenberg and quasi-triangular
*> in rows and columns INFO+1 through IHI.
*>
*> If INFO .GT. 0 and COMPZ = 'V', then on exit
*>
*> (final value of Z) = (initial value of Z)*U
*>
*> where U is the orthogonal matrix in (*) (regard-
*> less of the value of JOB.)
*>
*> If INFO .GT. 0 and COMPZ = 'I', then on exit
*> (final value of Z) = U
*> where U is the orthogonal matrix in (*) (regard-
*> less of the value of JOB.)
*>
*> If INFO .GT. 0 and COMPZ = 'N', then Z is not
*> accessed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Default values supplied by
*> ILAENV(ISPEC,'DHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
*> It is suggested that these defaults be adjusted in order
*> to attain best performance in each particular
*> computational environment.
*>
*> ISPEC=12: The DLAHQR vs DLAQR0 crossover point.
*> Default: 75. (Must be at least 11.)
*>
*> ISPEC=13: Recommended deflation window size.
*> This depends on ILO, IHI and NS. NS is the
*> number of simultaneous shifts returned
*> by ILAENV(ISPEC=15). (See ISPEC=15 below.)
*> The default for (IHI-ILO+1).LE.500 is NS.
*> The default for (IHI-ILO+1).GT.500 is 3*NS/2.
*>
*> ISPEC=14: Nibble crossover point. (See IPARMQ for
*> details.) Default: 14% of deflation window
*> size.
*>
*> ISPEC=15: Number of simultaneous shifts in a multishift
*> QR iteration.
*>
*> If IHI-ILO+1 is ...
*>
*> greater than ...but less ... the
*> or equal to ... than default is
*>
*> 1 30 NS = 2(+)
*> 30 60 NS = 4(+)
*> 60 150 NS = 10(+)
*> 150 590 NS = **
*> 590 3000 NS = 64
*> 3000 6000 NS = 128
*> 6000 infinity NS = 256
*>
*> (+) By default some or all matrices of this order
*> are passed to the implicit double shift routine
*> DLAHQR and this parameter is ignored. See
*> ISPEC=12 above and comments in IPARMQ for
*> details.
*>
*> (**) The asterisks (**) indicate an ad-hoc
*> function of N increasing from 10 to 64.
*>
*> ISPEC=16: Select structured matrix multiply.
*> If the number of simultaneous shifts (specified
*> by ISPEC=15) is less than 14, then the default
*> for ISPEC=16 is 0. Otherwise the default for
*> ISPEC=16 is 2.
*> \endverbatim
*
*> \par References:
* ================
*>
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
*> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
*> 929--947, 2002.
*> \n
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part II: Aggressive Early Deflation, SIAM Journal
*> of Matrix Analysis, volume 23, pages 948--973, 2002.
*
* =====================================================================
SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
$ LDZ, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
CHARACTER COMPZ, JOB
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
*
* ==== Matrices of order NTINY or smaller must be processed by
* . DLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
*
* ==== NL allocates some local workspace to help small matrices
* . through a rare DLAHQR failure. NL .GT. NTINY = 11 is
* . required and NL .LE. NMIN = ILAENV(ISPEC=12,...) is recom-
* . mended. (The default value of NMIN is 75.) Using NL = 49
* . allows up to six simultaneous shifts and a 16-by-16
* . deflation window. ====
INTEGER NL
PARAMETER ( NL = 49 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
* ..
* .. Local Arrays ..
DOUBLE PRECISION HL( NL, NL ), WORKL( NL )
* ..
* .. Local Scalars ..
INTEGER I, KBOT, NMIN
LOGICAL INITZ, LQUERY, WANTT, WANTZ
* ..
* .. External Functions ..
INTEGER ILAENV
LOGICAL LSAME
EXTERNAL ILAENV, LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DLAHQR, DLAQR0, DLASET, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN
* ..
* .. Executable Statements ..
*
* ==== Decode and check the input parameters. ====
*
WANTT = LSAME( JOB, 'S' )
INITZ = LSAME( COMPZ, 'I' )
WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
WORK( 1 ) = DBLE( MAX( 1, N ) )
LQUERY = LWORK.EQ.-1
*
INFO = 0
IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -5
ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
INFO = -11
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
*
IF( INFO.NE.0 ) THEN
*
* ==== Quick return in case of invalid argument. ====
*
CALL XERBLA( 'DHSEQR', -INFO )
RETURN
*
ELSE IF( N.EQ.0 ) THEN
*
* ==== Quick return in case N = 0; nothing to do. ====
*
RETURN
*
ELSE IF( LQUERY ) THEN
*
* ==== Quick return in case of a workspace query ====
*
CALL DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
$ IHI, Z, LDZ, WORK, LWORK, INFO )
* ==== Ensure reported workspace size is backward-compatible with
* . previous LAPACK versions. ====
WORK( 1 ) = MAX( DBLE( MAX( 1, N ) ), WORK( 1 ) )
RETURN
*
ELSE
*
* ==== copy eigenvalues isolated by DGEBAL ====
*
DO 10 I = 1, ILO - 1
WR( I ) = H( I, I )
WI( I ) = ZERO
10 CONTINUE
DO 20 I = IHI + 1, N
WR( I ) = H( I, I )
WI( I ) = ZERO
20 CONTINUE
*
* ==== Initialize Z, if requested ====
*
IF( INITZ )
$ CALL DLASET( 'A', N, N, ZERO, ONE, Z, LDZ )
*
* ==== Quick return if possible ====
*
IF( ILO.EQ.IHI ) THEN
WR( ILO ) = H( ILO, ILO )
WI( ILO ) = ZERO
RETURN
END IF
*
* ==== DLAHQR/DLAQR0 crossover point ====
*
NMIN = ILAENV( 12, 'DHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N,
$ ILO, IHI, LWORK )
NMIN = MAX( NTINY, NMIN )
*
* ==== DLAQR0 for big matrices; DLAHQR for small ones ====
*
IF( N.GT.NMIN ) THEN
CALL DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
$ IHI, Z, LDZ, WORK, LWORK, INFO )
ELSE
*
* ==== Small matrix ====
*
CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
$ IHI, Z, LDZ, INFO )
*
IF( INFO.GT.0 ) THEN
*
* ==== A rare DLAHQR failure! DLAQR0 sometimes succeeds
* . when DLAHQR fails. ====
*
KBOT = INFO
*
IF( N.GE.NL ) THEN
*
* ==== Larger matrices have enough subdiagonal scratch
* . space to call DLAQR0 directly. ====
*
CALL DLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, WR,
$ WI, ILO, IHI, Z, LDZ, WORK, LWORK, INFO )
*
ELSE
*
* ==== Tiny matrices don't have enough subdiagonal
* . scratch space to benefit from DLAQR0. Hence,
* . tiny matrices must be copied into a larger
* . array before calling DLAQR0. ====
*
CALL DLACPY( 'A', N, N, H, LDH, HL, NL )
HL( N+1, N ) = ZERO
CALL DLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ),
$ NL )
CALL DLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, WR,
$ WI, ILO, IHI, Z, LDZ, WORKL, NL, INFO )
IF( WANTT .OR. INFO.NE.0 )
$ CALL DLACPY( 'A', N, N, HL, NL, H, LDH )
END IF
END IF
END IF
*
* ==== Clear out the trash, if necessary. ====
*
IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 )
$ CALL DLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH )
*
* ==== Ensure reported workspace size is backward-compatible with
* . previous LAPACK versions. ====
*
WORK( 1 ) = MAX( DBLE( MAX( 1, N ) ), WORK( 1 ) )
END IF
*
* ==== End of DHSEQR ====
*
END
*> \brief \b DISNAN tests input for NaN.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DISNAN + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* LOGICAL FUNCTION DISNAN( DIN )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION DIN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DISNAN returns .TRUE. if its argument is NaN, and .FALSE.
*> otherwise. To be replaced by the Fortran 2003 intrinsic in the
*> future.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] DIN
*> \verbatim
*> DIN is DOUBLE PRECISION
*> Input to test for NaN.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
LOGICAL FUNCTION DISNAN( DIN )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
DOUBLE PRECISION DIN
* ..
*
* =====================================================================
*
* .. External Functions ..
LOGICAL DLAISNAN
EXTERNAL DLAISNAN
* ..
* .. Executable Statements ..
DISNAN = DLAISNAN(DIN,DIN)
RETURN
END
*> \brief \b DLABAD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLABAD + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLABAD( SMALL, LARGE )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION LARGE, SMALL
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLABAD takes as input the values computed by DLAMCH for underflow and
*> overflow, and returns the square root of each of these values if the
*> log of LARGE is sufficiently large. This subroutine is intended to
*> identify machines with a large exponent range, such as the Crays, and
*> redefine the underflow and overflow limits to be the square roots of
*> the values computed by DLAMCH. This subroutine is needed because
*> DLAMCH does not compensate for poor arithmetic in the upper half of
*> the exponent range, as is found on a Cray.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in,out] SMALL
*> \verbatim
*> SMALL is DOUBLE PRECISION
*> On entry, the underflow threshold as computed by DLAMCH.
*> On exit, if LOG10(LARGE) is sufficiently large, the square
*> root of SMALL, otherwise unchanged.
*> \endverbatim
*>
*> \param[in,out] LARGE
*> \verbatim
*> LARGE is DOUBLE PRECISION
*> On entry, the overflow threshold as computed by DLAMCH.
*> On exit, if LOG10(LARGE) is sufficiently large, the square
*> root of LARGE, otherwise unchanged.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLABAD( SMALL, LARGE )
*
* -- LAPACK auxiliary routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
DOUBLE PRECISION LARGE, SMALL
* ..
*
* =====================================================================
*
* .. Intrinsic Functions ..
INTRINSIC LOG10, SQRT
* ..
* .. Executable Statements ..
*
* If it looks like we're on a Cray, take the square root of
* SMALL and LARGE to avoid overflow and underflow problems.
*
IF( LOG10( LARGE ).GT.2000.D0 ) THEN
SMALL = SQRT( SMALL )
LARGE = SQRT( LARGE )
END IF
*
RETURN
*
* End of DLABAD
*
END
*> \brief \b DLACPY copies all or part of one two-dimensional array to another.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLACPY + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLACPY( UPLO, M, N, A, LDA, B, LDB )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDA, LDB, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLACPY copies all or part of a two-dimensional matrix A to another
*> matrix B.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies the part of the matrix A to be copied to B.
*> = 'U': Upper triangular part
*> = 'L': Lower triangular part
*> Otherwise: All of the matrix A
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The m by n matrix A. If UPLO = 'U', only the upper triangle
*> or trapezoid is accessed; if UPLO = 'L', only the lower
*> triangle or trapezoid is accessed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On exit, B = A in the locations specified by UPLO.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLACPY( UPLO, M, N, A, LDA, B, LDB )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDB, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( LSAME( UPLO, 'U' ) ) THEN
DO 20 J = 1, N
DO 10 I = 1, MIN( J, M )
B( I, J ) = A( I, J )
10 CONTINUE
20 CONTINUE
ELSE IF( LSAME( UPLO, 'L' ) ) THEN
DO 40 J = 1, N
DO 30 I = J, M
B( I, J ) = A( I, J )
30 CONTINUE
40 CONTINUE
ELSE
DO 60 J = 1, N
DO 50 I = 1, M
B( I, J ) = A( I, J )
50 CONTINUE
60 CONTINUE
END IF
RETURN
*
* End of DLACPY
*
END
*> \brief \b DLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLADIV + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLADIV( A, B, C, D, P, Q )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION A, B, C, D, P, Q
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLADIV performs complex division in real arithmetic
*>
*> a + i*b
*> p + i*q = ---------
*> c + i*d
*>
*> The algorithm is due to Michael Baudin and Robert L. Smith
*> and can be found in the paper
*> "A Robust Complex Division in Scilab"
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION
*> The scalars a, b, c, and d in the above expression.
*> \endverbatim
*>
*> \param[out] P
*> \verbatim
*> P is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION
*> The scalars p and q in the above expression.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date January 2013
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLADIV( A, B, C, D, P, Q )
*
* -- LAPACK auxiliary routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* January 2013
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B, C, D, P, Q
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION BS
PARAMETER ( BS = 2.0D0 )
DOUBLE PRECISION HALF
PARAMETER ( HALF = 0.5D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
*
* .. Local Scalars ..
DOUBLE PRECISION AA, BB, CC, DD, AB, CD, S, OV, UN, BE, EPS
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLADIV1
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
AA = A
BB = B
CC = C
DD = D
AB = MAX( ABS(A), ABS(B) )
CD = MAX( ABS(C), ABS(D) )
S = 1.0D0
OV = DLAMCH( 'Overflow threshold' )
UN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Epsilon' )
BE = BS / (EPS*EPS)
IF( AB >= HALF*OV ) THEN
AA = HALF * AA
BB = HALF * BB
S = TWO * S
END IF
IF( CD >= HALF*OV ) THEN
CC = HALF * CC
DD = HALF * DD
S = HALF * S
END IF
IF( AB <= UN*BS/EPS ) THEN
AA = AA * BE
BB = BB * BE
S = S / BE
END IF
IF( CD <= UN*BS/EPS ) THEN
CC = CC * BE
DD = DD * BE
S = S * BE
END IF
IF( ABS( D ).LE.ABS( C ) ) THEN
CALL DLADIV1(AA, BB, CC, DD, P, Q)
ELSE
CALL DLADIV1(BB, AA, DD, CC, P, Q)
Q = -Q
END IF
P = P * S
Q = Q * S
*
RETURN
*
* End of DLADIV
*
END
SUBROUTINE DLADIV1( A, B, C, D, P, Q )
*
* -- LAPACK auxiliary routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* January 2013
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B, C, D, P, Q
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
*
* .. Local Scalars ..
DOUBLE PRECISION R, T
* ..
* .. External Functions ..
DOUBLE PRECISION DLADIV2
EXTERNAL DLADIV2
* ..
* .. Executable Statements ..
*
R = D / C
T = ONE / (C + D * R)
P = DLADIV2(A, B, C, D, R, T)
A = -A
Q = DLADIV2(B, A, C, D, R, T)
*
RETURN
*
* End of DLADIV1
*
END
DOUBLE PRECISION FUNCTION DLADIV2( A, B, C, D, R, T )
*
* -- LAPACK auxiliary routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* January 2013
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B, C, D, R, T
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
*
* .. Local Scalars ..
DOUBLE PRECISION BR
* ..
* .. Executable Statements ..
*
IF( R.NE.ZERO ) THEN
BR = B * R
IF( BR.NE.ZERO ) THEN
DLADIV2 = (A + BR) * T
ELSE
DLADIV2 = A * T + (B * T) * R
END IF
ELSE
DLADIV2 = (A + D * (B / C)) * T
END IF
*
RETURN
*
* End of DLADIV12
*
END
*> \brief \b DLAEXC swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAEXC + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAEXC( WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK,
* INFO )
*
* .. Scalar Arguments ..
* LOGICAL WANTQ
* INTEGER INFO, J1, LDQ, LDT, N, N1, N2
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in
*> an upper quasi-triangular matrix T by an orthogonal similarity
*> transformation.
*>
*> T must be in Schur canonical form, that is, block upper triangular
*> with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block
*> has its diagonal elemnts equal and its off-diagonal elements of
*> opposite sign.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTQ
*> \verbatim
*> WANTQ is LOGICAL
*> = .TRUE. : accumulate the transformation in the matrix Q;
*> = .FALSE.: do not accumulate the transformation.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in,out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> On entry, the upper quasi-triangular matrix T, in Schur
*> canonical form.
*> On exit, the updated matrix T, again in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> On entry, if WANTQ is .TRUE., the orthogonal matrix Q.
*> On exit, if WANTQ is .TRUE., the updated matrix Q.
*> If WANTQ is .FALSE., Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q.
*> LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N.
*> \endverbatim
*>
*> \param[in] J1
*> \verbatim
*> J1 is INTEGER
*> The index of the first row of the first block T11.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> The order of the first block T11. N1 = 0, 1 or 2.
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*> N2 is INTEGER
*> The order of the second block T22. N2 = 0, 1 or 2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> = 1: the transformed matrix T would be too far from Schur
*> form; the blocks are not swapped and T and Q are
*> unchanged.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLAEXC( WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK,
$ INFO )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
LOGICAL WANTQ
INTEGER INFO, J1, LDQ, LDT, N, N1, N2
* ..
* .. Array Arguments ..
DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION TEN
PARAMETER ( TEN = 1.0D+1 )
INTEGER LDD, LDX
PARAMETER ( LDD = 4, LDX = 2 )
* ..
* .. Local Scalars ..
INTEGER IERR, J2, J3, J4, K, ND
DOUBLE PRECISION CS, DNORM, EPS, SCALE, SMLNUM, SN, T11, T22,
$ T33, TAU, TAU1, TAU2, TEMP, THRESH, WI1, WI2,
$ WR1, WR2, XNORM
* ..
* .. Local Arrays ..
DOUBLE PRECISION D( LDD, 4 ), U( 3 ), U1( 3 ), U2( 3 ),
$ X( LDX, 2 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL DLAMCH, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DLANV2, DLARFG, DLARFX, DLARTG, DLASY2,
$ DROT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.EQ.0 .OR. N1.EQ.0 .OR. N2.EQ.0 )
$ RETURN
IF( J1+N1.GT.N )
$ RETURN
*
J2 = J1 + 1
J3 = J1 + 2
J4 = J1 + 3
*
IF( N1.EQ.1 .AND. N2.EQ.1 ) THEN
*
* Swap two 1-by-1 blocks.
*
T11 = T( J1, J1 )
T22 = T( J2, J2 )
*
* Determine the transformation to perform the interchange.
*
CALL DLARTG( T( J1, J2 ), T22-T11, CS, SN, TEMP )
*
* Apply transformation to the matrix T.
*
IF( J3.LE.N )
$ CALL DROT( N-J1-1, T( J1, J3 ), LDT, T( J2, J3 ), LDT, CS,
$ SN )
CALL DROT( J1-1, T( 1, J1 ), 1, T( 1, J2 ), 1, CS, SN )
*
T( J1, J1 ) = T22
T( J2, J2 ) = T11
*
IF( WANTQ ) THEN
*
* Accumulate transformation in the matrix Q.
*
CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J2 ), 1, CS, SN )
END IF
*
ELSE
*
* Swapping involves at least one 2-by-2 block.
*
* Copy the diagonal block of order N1+N2 to the local array D
* and compute its norm.
*
ND = N1 + N2
CALL DLACPY( 'Full', ND, ND, T( J1, J1 ), LDT, D, LDD )
DNORM = DLANGE( 'Max', ND, ND, D, LDD, WORK )
*
* Compute machine-dependent threshold for test for accepting
* swap.
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
THRESH = MAX( TEN*EPS*DNORM, SMLNUM )
*
* Solve T11*X - X*T22 = scale*T12 for X.
*
CALL DLASY2( .FALSE., .FALSE., -1, N1, N2, D, LDD,
$ D( N1+1, N1+1 ), LDD, D( 1, N1+1 ), LDD, SCALE, X,
$ LDX, XNORM, IERR )
*
* Swap the adjacent diagonal blocks.
*
K = N1 + N1 + N2 - 3
GO TO ( 10, 20, 30 )K
*
10 CONTINUE
*
* N1 = 1, N2 = 2: generate elementary reflector H so that:
*
* ( scale, X11, X12 ) H = ( 0, 0, * )
*
U( 1 ) = SCALE
U( 2 ) = X( 1, 1 )
U( 3 ) = X( 1, 2 )
CALL DLARFG( 3, U( 3 ), U, 1, TAU )
U( 3 ) = ONE
T11 = T( J1, J1 )
*
* Perform swap provisionally on diagonal block in D.
*
CALL DLARFX( 'L', 3, 3, U, TAU, D, LDD, WORK )
CALL DLARFX( 'R', 3, 3, U, TAU, D, LDD, WORK )
*
* Test whether to reject swap.
*
IF( MAX( ABS( D( 3, 1 ) ), ABS( D( 3, 2 ) ), ABS( D( 3,
$ 3 )-T11 ) ).GT.THRESH )GO TO 50
*
* Accept swap: apply transformation to the entire matrix T.
*
CALL DLARFX( 'L', 3, N-J1+1, U, TAU, T( J1, J1 ), LDT, WORK )
CALL DLARFX( 'R', J2, 3, U, TAU, T( 1, J1 ), LDT, WORK )
*
T( J3, J1 ) = ZERO
T( J3, J2 ) = ZERO
T( J3, J3 ) = T11
*
IF( WANTQ ) THEN
*
* Accumulate transformation in the matrix Q.
*
CALL DLARFX( 'R', N, 3, U, TAU, Q( 1, J1 ), LDQ, WORK )
END IF
GO TO 40
*
20 CONTINUE
*
* N1 = 2, N2 = 1: generate elementary reflector H so that:
*
* H ( -X11 ) = ( * )
* ( -X21 ) = ( 0 )
* ( scale ) = ( 0 )
*
U( 1 ) = -X( 1, 1 )
U( 2 ) = -X( 2, 1 )
U( 3 ) = SCALE
CALL DLARFG( 3, U( 1 ), U( 2 ), 1, TAU )
U( 1 ) = ONE
T33 = T( J3, J3 )
*
* Perform swap provisionally on diagonal block in D.
*
CALL DLARFX( 'L', 3, 3, U, TAU, D, LDD, WORK )
CALL DLARFX( 'R', 3, 3, U, TAU, D, LDD, WORK )
*
* Test whether to reject swap.
*
IF( MAX( ABS( D( 2, 1 ) ), ABS( D( 3, 1 ) ), ABS( D( 1,
$ 1 )-T33 ) ).GT.THRESH )GO TO 50
*
* Accept swap: apply transformation to the entire matrix T.
*
CALL DLARFX( 'R', J3, 3, U, TAU, T( 1, J1 ), LDT, WORK )
CALL DLARFX( 'L', 3, N-J1, U, TAU, T( J1, J2 ), LDT, WORK )
*
T( J1, J1 ) = T33
T( J2, J1 ) = ZERO
T( J3, J1 ) = ZERO
*
IF( WANTQ ) THEN
*
* Accumulate transformation in the matrix Q.
*
CALL DLARFX( 'R', N, 3, U, TAU, Q( 1, J1 ), LDQ, WORK )
END IF
GO TO 40
*
30 CONTINUE
*
* N1 = 2, N2 = 2: generate elementary reflectors H(1) and H(2) so
* that:
*
* H(2) H(1) ( -X11 -X12 ) = ( * * )
* ( -X21 -X22 ) ( 0 * )
* ( scale 0 ) ( 0 0 )
* ( 0 scale ) ( 0 0 )
*
U1( 1 ) = -X( 1, 1 )
U1( 2 ) = -X( 2, 1 )
U1( 3 ) = SCALE
CALL DLARFG( 3, U1( 1 ), U1( 2 ), 1, TAU1 )
U1( 1 ) = ONE
*
TEMP = -TAU1*( X( 1, 2 )+U1( 2 )*X( 2, 2 ) )
U2( 1 ) = -TEMP*U1( 2 ) - X( 2, 2 )
U2( 2 ) = -TEMP*U1( 3 )
U2( 3 ) = SCALE
CALL DLARFG( 3, U2( 1 ), U2( 2 ), 1, TAU2 )
U2( 1 ) = ONE
*
* Perform swap provisionally on diagonal block in D.
*
CALL DLARFX( 'L', 3, 4, U1, TAU1, D, LDD, WORK )
CALL DLARFX( 'R', 4, 3, U1, TAU1, D, LDD, WORK )
CALL DLARFX( 'L', 3, 4, U2, TAU2, D( 2, 1 ), LDD, WORK )
CALL DLARFX( 'R', 4, 3, U2, TAU2, D( 1, 2 ), LDD, WORK )
*
* Test whether to reject swap.
*
IF( MAX( ABS( D( 3, 1 ) ), ABS( D( 3, 2 ) ), ABS( D( 4, 1 ) ),
$ ABS( D( 4, 2 ) ) ).GT.THRESH )GO TO 50
*
* Accept swap: apply transformation to the entire matrix T.
*
CALL DLARFX( 'L', 3, N-J1+1, U1, TAU1, T( J1, J1 ), LDT, WORK )
CALL DLARFX( 'R', J4, 3, U1, TAU1, T( 1, J1 ), LDT, WORK )
CALL DLARFX( 'L', 3, N-J1+1, U2, TAU2, T( J2, J1 ), LDT, WORK )
CALL DLARFX( 'R', J4, 3, U2, TAU2, T( 1, J2 ), LDT, WORK )
*
T( J3, J1 ) = ZERO
T( J3, J2 ) = ZERO
T( J4, J1 ) = ZERO
T( J4, J2 ) = ZERO
*
IF( WANTQ ) THEN
*
* Accumulate transformation in the matrix Q.
*
CALL DLARFX( 'R', N, 3, U1, TAU1, Q( 1, J1 ), LDQ, WORK )
CALL DLARFX( 'R', N, 3, U2, TAU2, Q( 1, J2 ), LDQ, WORK )
END IF
*
40 CONTINUE
*
IF( N2.EQ.2 ) THEN
*
* Standardize new 2-by-2 block T11
*
CALL DLANV2( T( J1, J1 ), T( J1, J2 ), T( J2, J1 ),
$ T( J2, J2 ), WR1, WI1, WR2, WI2, CS, SN )
CALL DROT( N-J1-1, T( J1, J1+2 ), LDT, T( J2, J1+2 ), LDT,
$ CS, SN )
CALL DROT( J1-1, T( 1, J1 ), 1, T( 1, J2 ), 1, CS, SN )
IF( WANTQ )
$ CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J2 ), 1, CS, SN )
END IF
*
IF( N1.EQ.2 ) THEN
*
* Standardize new 2-by-2 block T22
*
J3 = J1 + N2
J4 = J3 + 1
CALL DLANV2( T( J3, J3 ), T( J3, J4 ), T( J4, J3 ),
$ T( J4, J4 ), WR1, WI1, WR2, WI2, CS, SN )
IF( J3+2.LE.N )
$ CALL DROT( N-J3-1, T( J3, J3+2 ), LDT, T( J4, J3+2 ),
$ LDT, CS, SN )
CALL DROT( J3-1, T( 1, J3 ), 1, T( 1, J4 ), 1, CS, SN )
IF( WANTQ )
$ CALL DROT( N, Q( 1, J3 ), 1, Q( 1, J4 ), 1, CS, SN )
END IF
*
END IF
RETURN
*
* Exit with INFO = 1 if swap was rejected.
*
50 CONTINUE
INFO = 1
RETURN
*
* End of DLAEXC
*
END
*> \brief \b DLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAHQR + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
* ILOZ, IHIZ, Z, LDZ, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAHQR is an auxiliary routine called by DHSEQR to update the
*> eigenvalues and Schur decomposition already computed by DHSEQR, by
*> dealing with the Hessenberg submatrix in rows and columns ILO to
*> IHI.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> = .TRUE. : the full Schur form T is required;
*> = .FALSE.: only eigenvalues are required.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> = .TRUE. : the matrix of Schur vectors Z is required;
*> = .FALSE.: Schur vectors are not required.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> It is assumed that H is already upper quasi-triangular in
*> rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
*> ILO = 1). DLAHQR works primarily with the Hessenberg
*> submatrix in rows and columns ILO to IHI, but applies
*> transformations to all of H if WANTT is .TRUE..
*> 1 <= ILO <= max(1,IHI); IHI <= N.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On entry, the upper Hessenberg matrix H.
*> On exit, if INFO is zero and if WANTT is .TRUE., H is upper
*> quasi-triangular in rows and columns ILO:IHI, with any
*> 2-by-2 diagonal blocks in standard form. If INFO is zero
*> and WANTT is .FALSE., the contents of H are unspecified on
*> exit. The output state of H if INFO is nonzero is given
*> below under the description of INFO.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (N)
*> The real and imaginary parts, respectively, of the computed
*> eigenvalues ILO to IHI are stored in the corresponding
*> elements of WR and WI. If two eigenvalues are computed as a
*> complex conjugate pair, they are stored in consecutive
*> elements of WR and WI, say the i-th and (i+1)th, with
*> WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
*> eigenvalues are stored in the same order as on the diagonal
*> of the Schur form returned in H, with WR(i) = H(i,i), and, if
*> H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
*> WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE..
*> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> If WANTZ is .TRUE., on entry Z must contain the current
*> matrix Z of transformations accumulated by DHSEQR, and on
*> exit Z has been updated; transformations are applied only to
*> the submatrix Z(ILOZ:IHIZ,ILO:IHI).
*> If WANTZ is .FALSE., Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> .GT. 0: If INFO = i, DLAHQR failed to compute all the
*> eigenvalues ILO to IHI in a total of 30 iterations
*> per eigenvalue; elements i+1:ihi of WR and WI
*> contain those eigenvalues which have been
*> successfully computed.
*>
*> If INFO .GT. 0 and WANTT is .FALSE., then on exit,
*> the remaining unconverged eigenvalues are the
*> eigenvalues of the upper Hessenberg matrix rows
*> and columns ILO thorugh INFO of the final, output
*> value of H.
*>
*> If INFO .GT. 0 and WANTT is .TRUE., then on exit
*> (*) (initial value of H)*U = U*(final value of H)
*> where U is an orthognal matrix. The final
*> value of H is upper Hessenberg and triangular in
*> rows and columns INFO+1 through IHI.
*>
*> If INFO .GT. 0 and WANTZ is .TRUE., then on exit
*> (final value of Z) = (initial value of Z)*U
*> where U is the orthogonal matrix in (*)
*> (regardless of the value of WANTT.)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2015
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> 02-96 Based on modifications by
*> David Day, Sandia National Laboratory, USA
*>
*> 12-04 Further modifications by
*> Ralph Byers, University of Kansas, USA
*> This is a modified version of DLAHQR from LAPACK version 3.0.
*> It is (1) more robust against overflow and underflow and
*> (2) adopts the more conservative Ahues & Tisseur stopping
*> criterion (LAWN 122, 1997).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, INFO )
*
* -- LAPACK auxiliary routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2015
*
* .. Scalar Arguments ..
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
* ..
*
* =========================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0, TWO = 2.0d0 )
DOUBLE PRECISION DAT1, DAT2
PARAMETER ( DAT1 = 3.0d0 / 4.0d0, DAT2 = -0.4375d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, AB, BA, BB, CS, DET, H11, H12, H21, H21S,
$ H22, RT1I, RT1R, RT2I, RT2R, RTDISC, S, SAFMAX,
$ SAFMIN, SMLNUM, SN, SUM, T1, T2, T3, TR, TST,
$ ULP, V2, V3
INTEGER I, I1, I2, ITS, ITMAX, J, K, L, M, NH, NR, NZ
* ..
* .. Local Arrays ..
DOUBLE PRECISION V( 3 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLABAD, DLANV2, DLARFG, DROT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
IF( ILO.EQ.IHI ) THEN
WR( ILO ) = H( ILO, ILO )
WI( ILO ) = ZERO
RETURN
END IF
*
* ==== clear out the trash ====
DO 10 J = ILO, IHI - 3
H( J+2, J ) = ZERO
H( J+3, J ) = ZERO
10 CONTINUE
IF( ILO.LE.IHI-2 )
$ H( IHI, IHI-2 ) = ZERO
*
NH = IHI - ILO + 1
NZ = IHIZ - ILOZ + 1
*
* Set machine-dependent constants for the stopping criterion.
*
SAFMIN = DLAMCH( 'SAFE MINIMUM' )
SAFMAX = ONE / SAFMIN
CALL DLABAD( SAFMIN, SAFMAX )
ULP = DLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( DBLE( NH ) / ULP )
*
* I1 and I2 are the indices of the first row and last column of H
* to which transformations must be applied. If eigenvalues only are
* being computed, I1 and I2 are set inside the main loop.
*
IF( WANTT ) THEN
I1 = 1
I2 = N
END IF
*
* ITMAX is the total number of QR iterations allowed.
*
ITMAX = 30 * MAX( 10, NH )
*
* The main loop begins here. I is the loop index and decreases from
* IHI to ILO in steps of 1 or 2. Each iteration of the loop works
* with the active submatrix in rows and columns L to I.
* Eigenvalues I+1 to IHI have already converged. Either L = ILO or
* H(L,L-1) is negligible so that the matrix splits.
*
I = IHI
20 CONTINUE
L = ILO
IF( I.LT.ILO )
$ GO TO 160
*
* Perform QR iterations on rows and columns ILO to I until a
* submatrix of order 1 or 2 splits off at the bottom because a
* subdiagonal element has become negligible.
*
DO 140 ITS = 0, ITMAX
*
* Look for a single small subdiagonal element.
*
DO 30 K = I, L + 1, -1
IF( ABS( H( K, K-1 ) ).LE.SMLNUM )
$ GO TO 40
TST = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) )
IF( TST.EQ.ZERO ) THEN
IF( K-2.GE.ILO )
$ TST = TST + ABS( H( K-1, K-2 ) )
IF( K+1.LE.IHI )
$ TST = TST + ABS( H( K+1, K ) )
END IF
* ==== The following is a conservative small subdiagonal
* . deflation criterion due to Ahues & Tisseur (LAWN 122,
* . 1997). It has better mathematical foundation and
* . improves accuracy in some cases. ====
IF( ABS( H( K, K-1 ) ).LE.ULP*TST ) THEN
AB = MAX( ABS( H( K, K-1 ) ), ABS( H( K-1, K ) ) )
BA = MIN( ABS( H( K, K-1 ) ), ABS( H( K-1, K ) ) )
AA = MAX( ABS( H( K, K ) ),
$ ABS( H( K-1, K-1 )-H( K, K ) ) )
BB = MIN( ABS( H( K, K ) ),
$ ABS( H( K-1, K-1 )-H( K, K ) ) )
S = AA + AB
IF( BA*( AB / S ).LE.MAX( SMLNUM,
$ ULP*( BB*( AA / S ) ) ) )GO TO 40
END IF
30 CONTINUE
40 CONTINUE
L = K
IF( L.GT.ILO ) THEN
*
* H(L,L-1) is negligible
*
H( L, L-1 ) = ZERO
END IF
*
* Exit from loop if a submatrix of order 1 or 2 has split off.
*
IF( L.GE.I-1 )
$ GO TO 150
*
* Now the active submatrix is in rows and columns L to I. If
* eigenvalues only are being computed, only the active submatrix
* need be transformed.
*
IF( .NOT.WANTT ) THEN
I1 = L
I2 = I
END IF
*
IF( ITS.EQ.10 ) THEN
*
* Exceptional shift.
*
S = ABS( H( L+1, L ) ) + ABS( H( L+2, L+1 ) )
H11 = DAT1*S + H( L, L )
H12 = DAT2*S
H21 = S
H22 = H11
ELSE IF( ITS.EQ.20 ) THEN
*
* Exceptional shift.
*
S = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
H11 = DAT1*S + H( I, I )
H12 = DAT2*S
H21 = S
H22 = H11
ELSE
*
* Prepare to use Francis' double shift
* (i.e. 2nd degree generalized Rayleigh quotient)
*
H11 = H( I-1, I-1 )
H21 = H( I, I-1 )
H12 = H( I-1, I )
H22 = H( I, I )
END IF
S = ABS( H11 ) + ABS( H12 ) + ABS( H21 ) + ABS( H22 )
IF( S.EQ.ZERO ) THEN
RT1R = ZERO
RT1I = ZERO
RT2R = ZERO
RT2I = ZERO
ELSE
H11 = H11 / S
H21 = H21 / S
H12 = H12 / S
H22 = H22 / S
TR = ( H11+H22 ) / TWO
DET = ( H11-TR )*( H22-TR ) - H12*H21
RTDISC = SQRT( ABS( DET ) )
IF( DET.GE.ZERO ) THEN
*
* ==== complex conjugate shifts ====
*
RT1R = TR*S
RT2R = RT1R
RT1I = RTDISC*S
RT2I = -RT1I
ELSE
*
* ==== real shifts (use only one of them) ====
*
RT1R = TR + RTDISC
RT2R = TR - RTDISC
IF( ABS( RT1R-H22 ).LE.ABS( RT2R-H22 ) ) THEN
RT1R = RT1R*S
RT2R = RT1R
ELSE
RT2R = RT2R*S
RT1R = RT2R
END IF
RT1I = ZERO
RT2I = ZERO
END IF
END IF
*
* Look for two consecutive small subdiagonal elements.
*
DO 50 M = I - 2, L, -1
* Determine the effect of starting the double-shift QR
* iteration at row M, and see if this would make H(M,M-1)
* negligible. (The following uses scaling to avoid
* overflows and most underflows.)
*
H21S = H( M+1, M )
S = ABS( H( M, M )-RT2R ) + ABS( RT2I ) + ABS( H21S )
H21S = H( M+1, M ) / S
V( 1 ) = H21S*H( M, M+1 ) + ( H( M, M )-RT1R )*
$ ( ( H( M, M )-RT2R ) / S ) - RT1I*( RT2I / S )
V( 2 ) = H21S*( H( M, M )+H( M+1, M+1 )-RT1R-RT2R )
V( 3 ) = H21S*H( M+2, M+1 )
S = ABS( V( 1 ) ) + ABS( V( 2 ) ) + ABS( V( 3 ) )
V( 1 ) = V( 1 ) / S
V( 2 ) = V( 2 ) / S
V( 3 ) = V( 3 ) / S
IF( M.EQ.L )
$ GO TO 60
IF( ABS( H( M, M-1 ) )*( ABS( V( 2 ) )+ABS( V( 3 ) ) ).LE.
$ ULP*ABS( V( 1 ) )*( ABS( H( M-1, M-1 ) )+ABS( H( M,
$ M ) )+ABS( H( M+1, M+1 ) ) ) )GO TO 60
50 CONTINUE
60 CONTINUE
*
* Double-shift QR step
*
DO 130 K = M, I - 1
*
* The first iteration of this loop determines a reflection G
* from the vector V and applies it from left and right to H,
* thus creating a nonzero bulge below the subdiagonal.
*
* Each subsequent iteration determines a reflection G to
* restore the Hessenberg form in the (K-1)th column, and thus
* chases the bulge one step toward the bottom of the active
* submatrix. NR is the order of G.
*
NR = MIN( 3, I-K+1 )
IF( K.GT.M )
$ CALL DCOPY( NR, H( K, K-1 ), 1, V, 1 )
CALL DLARFG( NR, V( 1 ), V( 2 ), 1, T1 )
IF( K.GT.M ) THEN
H( K, K-1 ) = V( 1 )
H( K+1, K-1 ) = ZERO
IF( K.LT.I-1 )
$ H( K+2, K-1 ) = ZERO
ELSE IF( M.GT.L ) THEN
* ==== Use the following instead of
* . H( K, K-1 ) = -H( K, K-1 ) to
* . avoid a bug when v(2) and v(3)
* . underflow. ====
H( K, K-1 ) = H( K, K-1 )*( ONE-T1 )
END IF
V2 = V( 2 )
T2 = T1*V2
IF( NR.EQ.3 ) THEN
V3 = V( 3 )
T3 = T1*V3
*
* Apply G from the left to transform the rows of the matrix
* in columns K to I2.
*
DO 70 J = K, I2
SUM = H( K, J ) + V2*H( K+1, J ) + V3*H( K+2, J )
H( K, J ) = H( K, J ) - SUM*T1
H( K+1, J ) = H( K+1, J ) - SUM*T2
H( K+2, J ) = H( K+2, J ) - SUM*T3
70 CONTINUE
*
* Apply G from the right to transform the columns of the
* matrix in rows I1 to min(K+3,I).
*
DO 80 J = I1, MIN( K+3, I )
SUM = H( J, K ) + V2*H( J, K+1 ) + V3*H( J, K+2 )
H( J, K ) = H( J, K ) - SUM*T1
H( J, K+1 ) = H( J, K+1 ) - SUM*T2
H( J, K+2 ) = H( J, K+2 ) - SUM*T3
80 CONTINUE
*
IF( WANTZ ) THEN
*
* Accumulate transformations in the matrix Z
*
DO 90 J = ILOZ, IHIZ
SUM = Z( J, K ) + V2*Z( J, K+1 ) + V3*Z( J, K+2 )
Z( J, K ) = Z( J, K ) - SUM*T1
Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2
Z( J, K+2 ) = Z( J, K+2 ) - SUM*T3
90 CONTINUE
END IF
ELSE IF( NR.EQ.2 ) THEN
*
* Apply G from the left to transform the rows of the matrix
* in columns K to I2.
*
DO 100 J = K, I2
SUM = H( K, J ) + V2*H( K+1, J )
H( K, J ) = H( K, J ) - SUM*T1
H( K+1, J ) = H( K+1, J ) - SUM*T2
100 CONTINUE
*
* Apply G from the right to transform the columns of the
* matrix in rows I1 to min(K+3,I).
*
DO 110 J = I1, I
SUM = H( J, K ) + V2*H( J, K+1 )
H( J, K ) = H( J, K ) - SUM*T1
H( J, K+1 ) = H( J, K+1 ) - SUM*T2
110 CONTINUE
*
IF( WANTZ ) THEN
*
* Accumulate transformations in the matrix Z
*
DO 120 J = ILOZ, IHIZ
SUM = Z( J, K ) + V2*Z( J, K+1 )
Z( J, K ) = Z( J, K ) - SUM*T1
Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2
120 CONTINUE
END IF
END IF
130 CONTINUE
*
140 CONTINUE
*
* Failure to converge in remaining number of iterations
*
INFO = I
RETURN
*
150 CONTINUE
*
IF( L.EQ.I ) THEN
*
* H(I,I-1) is negligible: one eigenvalue has converged.
*
WR( I ) = H( I, I )
WI( I ) = ZERO
ELSE IF( L.EQ.I-1 ) THEN
*
* H(I-1,I-2) is negligible: a pair of eigenvalues have converged.
*
* Transform the 2-by-2 submatrix to standard Schur form,
* and compute and store the eigenvalues.
*
CALL DLANV2( H( I-1, I-1 ), H( I-1, I ), H( I, I-1 ),
$ H( I, I ), WR( I-1 ), WI( I-1 ), WR( I ), WI( I ),
$ CS, SN )
*
IF( WANTT ) THEN
*
* Apply the transformation to the rest of H.
*
IF( I2.GT.I )
$ CALL DROT( I2-I, H( I-1, I+1 ), LDH, H( I, I+1 ), LDH,
$ CS, SN )
CALL DROT( I-I1-1, H( I1, I-1 ), 1, H( I1, I ), 1, CS, SN )
END IF
IF( WANTZ ) THEN
*
* Apply the transformation to Z.
*
CALL DROT( NZ, Z( ILOZ, I-1 ), 1, Z( ILOZ, I ), 1, CS, SN )
END IF
END IF
*
* return to start of the main loop with new value of I.
*
I = L - 1
GO TO 20
*
160 CONTINUE
RETURN
*
* End of DLAHQR
*
END
*> \brief \b DLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAHR2 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
* .. Scalar Arguments ..
* INTEGER K, LDA, LDT, LDY, N, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
* $ Y( LDY, NB )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)
*> matrix A so that elements below the k-th subdiagonal are zero. The
*> reduction is performed by an orthogonal similarity transformation
*> Q**T * A * Q. The routine returns the matrices V and T which determine
*> Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
*>
*> This is an auxiliary routine called by DGEHRD.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The offset for the reduction. Elements below the k-th
*> subdiagonal in the first NB columns are reduced to zero.
*> K < N.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The number of columns to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N-K+1)
*> On entry, the n-by-(n-k+1) general matrix A.
*> On exit, the elements on and above the k-th subdiagonal in
*> the first NB columns are overwritten with the corresponding
*> elements of the reduced matrix; the elements below the k-th
*> subdiagonal, with the array TAU, represent the matrix Q as a
*> product of elementary reflectors. The other columns of A are
*> unchanged. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (NB)
*> The scalar factors of the elementary reflectors. See Further
*> Details.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,NB)
*> The upper triangular matrix T.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= NB.
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*> Y is DOUBLE PRECISION array, dimension (LDY,NB)
*> The n-by-nb matrix Y.
*> \endverbatim
*>
*> \param[in] LDY
*> \verbatim
*> LDY is INTEGER
*> The leading dimension of the array Y. LDY >= N.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of nb elementary reflectors
*>
*> Q = H(1) H(2) . . . H(nb).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
*> A(i+k+1:n,i), and tau in TAU(i).
*>
*> The elements of the vectors v together form the (n-k+1)-by-nb matrix
*> V which is needed, with T and Y, to apply the transformation to the
*> unreduced part of the matrix, using an update of the form:
*> A := (I - V*T*V**T) * (A - Y*V**T).
*>
*> The contents of A on exit are illustrated by the following example
*> with n = 7, k = 3 and nb = 2:
*>
*> ( a a a a a )
*> ( a a a a a )
*> ( a a a a a )
*> ( h h a a a )
*> ( v1 h a a a )
*> ( v1 v2 a a a )
*> ( v1 v2 a a a )
*>
*> where a denotes an element of the original matrix A, h denotes a
*> modified element of the upper Hessenberg matrix H, and vi denotes an
*> element of the vector defining H(i).
*>
*> This subroutine is a slight modification of LAPACK-3.0's DLAHRD
*> incorporating improvements proposed by Quintana-Orti and Van de
*> Gejin. Note that the entries of A(1:K,2:NB) differ from those
*> returned by the original LAPACK-3.0's DLAHRD routine. (This
*> subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
*> \endverbatim
*
*> \par References:
* ================
*>
*> Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
*> performance of reduction to Hessenberg form," ACM Transactions on
*> Mathematical Software, 32(2):180-194, June 2006.
*>
* =====================================================================
SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER K, LDA, LDT, LDY, N, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
$ Y( LDY, NB )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0,
$ ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION EI
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMM, DGEMV, DLACPY,
$ DLARFG, DSCAL, DTRMM, DTRMV
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
DO 10 I = 1, NB
IF( I.GT.1 ) THEN
*
* Update A(K+1:N,I)
*
* Update I-th column of A - Y * V**T
*
CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
$ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
*
* Apply I - V * T**T * V**T to this column (call it b) from the
* left, using the last column of T as workspace
*
* Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
* ( V2 ) ( b2 )
*
* where V1 is unit lower triangular
*
* w := V1**T * b1
*
CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
CALL DTRMV( 'Lower', 'Transpose', 'UNIT',
$ I-1, A( K+1, 1 ),
$ LDA, T( 1, NB ), 1 )
*
* w := w + V2**T * b2
*
CALL DGEMV( 'Transpose', N-K-I+1, I-1,
$ ONE, A( K+I, 1 ),
$ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
*
* w := T**T * w
*
CALL DTRMV( 'Upper', 'Transpose', 'NON-UNIT',
$ I-1, T, LDT,
$ T( 1, NB ), 1 )
*
* b2 := b2 - V2*w
*
CALL DGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
$ A( K+I, 1 ),
$ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
*
* b1 := b1 - V1*w
*
CALL DTRMV( 'Lower', 'NO TRANSPOSE',
$ 'UNIT', I-1,
$ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
*
A( K+I-1, I-1 ) = EI
END IF
*
* Generate the elementary reflector H(I) to annihilate
* A(K+I+1:N,I)
*
CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
$ TAU( I ) )
EI = A( K+I, I )
A( K+I, I ) = ONE
*
* Compute Y(K+1:N,I)
*
CALL DGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
$ ONE, A( K+1, I+1 ),
$ LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
CALL DGEMV( 'Transpose', N-K-I+1, I-1,
$ ONE, A( K+I, 1 ), LDA,
$ A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
$ Y( K+1, 1 ), LDY,
$ T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
CALL DSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
*
* Compute T(1:I,I)
*
CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
CALL DTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
$ I-1, T, LDT,
$ T( 1, I ), 1 )
T( I, I ) = TAU( I )
*
10 CONTINUE
A( K+NB, NB ) = EI
*
* Compute Y(1:K,1:NB)
*
CALL DLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
CALL DTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
$ 'UNIT', K, NB,
$ ONE, A( K+1, 1 ), LDA, Y, LDY )
IF( N.GT.K+NB )
$ CALL DGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
$ NB, N-K-NB, ONE,
$ A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
$ LDY )
CALL DTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
$ 'NON-UNIT', K, NB,
$ ONE, T, LDT, Y, LDY )
*
RETURN
*
* End of DLAHR2
*
END
*> \brief \b DLAISNAN tests input for NaN by comparing two arguments for inequality.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAISNAN + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* LOGICAL FUNCTION DLAISNAN( DIN1, DIN2 )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION DIN1, DIN2
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is not for general use. It exists solely to avoid
*> over-optimization in DISNAN.
*>
*> DLAISNAN checks for NaNs by comparing its two arguments for
*> inequality. NaN is the only floating-point value where NaN != NaN
*> returns .TRUE. To check for NaNs, pass the same variable as both
*> arguments.
*>
*> A compiler must assume that the two arguments are
*> not the same variable, and the test will not be optimized away.
*> Interprocedural or whole-program optimization may delete this
*> test. The ISNAN functions will be replaced by the correct
*> Fortran 03 intrinsic once the intrinsic is widely available.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] DIN1
*> \verbatim
*> DIN1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] DIN2
*> \verbatim
*> DIN2 is DOUBLE PRECISION
*> Two numbers to compare for inequality.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
LOGICAL FUNCTION DLAISNAN( DIN1, DIN2 )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
DOUBLE PRECISION DIN1, DIN2
* ..
*
* =====================================================================
*
* .. Executable Statements ..
DLAISNAN = (DIN1.NE.DIN2)
RETURN
END
*> \brief \b DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLALN2 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
* LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
*
* .. Scalar Arguments ..
* LOGICAL LTRANS
* INTEGER INFO, LDA, LDB, LDX, NA, NW
* DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLALN2 solves a system of the form (ca A - w D ) X = s B
*> or (ca A**T - w D) X = s B with possible scaling ("s") and
*> perturbation of A. (A**T means A-transpose.)
*>
*> A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
*> real diagonal matrix, w is a real or complex value, and X and B are
*> NA x 1 matrices -- real if w is real, complex if w is complex. NA
*> may be 1 or 2.
*>
*> If w is complex, X and B are represented as NA x 2 matrices,
*> the first column of each being the real part and the second
*> being the imaginary part.
*>
*> "s" is a scaling factor (.LE. 1), computed by DLALN2, which is
*> so chosen that X can be computed without overflow. X is further
*> scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
*> than overflow.
*>
*> If both singular values of (ca A - w D) are less than SMIN,
*> SMIN*identity will be used instead of (ca A - w D). If only one
*> singular value is less than SMIN, one element of (ca A - w D) will be
*> perturbed enough to make the smallest singular value roughly SMIN.
*> If both singular values are at least SMIN, (ca A - w D) will not be
*> perturbed. In any case, the perturbation will be at most some small
*> multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values
*> are computed by infinity-norm approximations, and thus will only be
*> correct to a factor of 2 or so.
*>
*> Note: all input quantities are assumed to be smaller than overflow
*> by a reasonable factor. (See BIGNUM.)
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] LTRANS
*> \verbatim
*> LTRANS is LOGICAL
*> =.TRUE.: A-transpose will be used.
*> =.FALSE.: A will be used (not transposed.)
*> \endverbatim
*>
*> \param[in] NA
*> \verbatim
*> NA is INTEGER
*> The size of the matrix A. It may (only) be 1 or 2.
*> \endverbatim
*>
*> \param[in] NW
*> \verbatim
*> NW is INTEGER
*> 1 if "w" is real, 2 if "w" is complex. It may only be 1
*> or 2.
*> \endverbatim
*>
*> \param[in] SMIN
*> \verbatim
*> SMIN is DOUBLE PRECISION
*> The desired lower bound on the singular values of A. This
*> should be a safe distance away from underflow or overflow,
*> say, between (underflow/machine precision) and (machine
*> precision * overflow ). (See BIGNUM and ULP.)
*> \endverbatim
*>
*> \param[in] CA
*> \verbatim
*> CA is DOUBLE PRECISION
*> The coefficient c, which A is multiplied by.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,NA)
*> The NA x NA matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. It must be at least NA.
*> \endverbatim
*>
*> \param[in] D1
*> \verbatim
*> D1 is DOUBLE PRECISION
*> The 1,1 element in the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] D2
*> \verbatim
*> D2 is DOUBLE PRECISION
*> The 2,2 element in the diagonal matrix D. Not used if NW=1.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NW)
*> The NA x NW matrix B (right-hand side). If NW=2 ("w" is
*> complex), column 1 contains the real part of B and column 2
*> contains the imaginary part.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. It must be at least NA.
*> \endverbatim
*>
*> \param[in] WR
*> \verbatim
*> WR is DOUBLE PRECISION
*> The real part of the scalar "w".
*> \endverbatim
*>
*> \param[in] WI
*> \verbatim
*> WI is DOUBLE PRECISION
*> The imaginary part of the scalar "w". Not used if NW=1.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NW)
*> The NA x NW matrix X (unknowns), as computed by DLALN2.
*> If NW=2 ("w" is complex), on exit, column 1 will contain
*> the real part of X and column 2 will contain the imaginary
*> part.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of X. It must be at least NA.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> The scale factor that B must be multiplied by to insure
*> that overflow does not occur when computing X. Thus,
*> (ca A - w D) X will be SCALE*B, not B (ignoring
*> perturbations of A.) It will be at most 1.
*> \endverbatim
*>
*> \param[out] XNORM
*> \verbatim
*> XNORM is DOUBLE PRECISION
*> The infinity-norm of X, when X is regarded as an NA x NW
*> real matrix.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> An error flag. It will be set to zero if no error occurs,
*> a negative number if an argument is in error, or a positive
*> number if ca A - w D had to be perturbed.
*> The possible values are:
*> = 0: No error occurred, and (ca A - w D) did not have to be
*> perturbed.
*> = 1: (ca A - w D) had to be perturbed to make its smallest
*> (or only) singular value greater than SMIN.
*> NOTE: In the interests of speed, this routine does not
*> check the inputs for errors.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
$ LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
LOGICAL LTRANS
INTEGER INFO, LDA, LDB, LDX, NA, NW
DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
* ..
* .. Local Scalars ..
INTEGER ICMAX, J
DOUBLE PRECISION BBND, BI1, BI2, BIGNUM, BNORM, BR1, BR2, CI21,
$ CI22, CMAX, CNORM, CR21, CR22, CSI, CSR, LI21,
$ LR21, SMINI, SMLNUM, TEMP, U22ABS, UI11, UI11R,
$ UI12, UI12S, UI22, UR11, UR11R, UR12, UR12S,
$ UR22, XI1, XI2, XR1, XR2
* ..
* .. Local Arrays ..
LOGICAL RSWAP( 4 ), ZSWAP( 4 )
INTEGER IPIVOT( 4, 4 )
DOUBLE PRECISION CI( 2, 2 ), CIV( 4 ), CR( 2, 2 ), CRV( 4 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLADIV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Equivalences ..
EQUIVALENCE ( CI( 1, 1 ), CIV( 1 ) ),
$ ( CR( 1, 1 ), CRV( 1 ) )
* ..
* .. Data statements ..
DATA ZSWAP / .FALSE., .FALSE., .TRUE., .TRUE. /
DATA RSWAP / .FALSE., .TRUE., .FALSE., .TRUE. /
DATA IPIVOT / 1, 2, 3, 4, 2, 1, 4, 3, 3, 4, 1, 2, 4,
$ 3, 2, 1 /
* ..
* .. Executable Statements ..
*
* Compute BIGNUM
*
SMLNUM = TWO*DLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
SMINI = MAX( SMIN, SMLNUM )
*
* Don't check for input errors
*
INFO = 0
*
* Standard Initializations
*
SCALE = ONE
*
IF( NA.EQ.1 ) THEN
*
* 1 x 1 (i.e., scalar) system C X = B
*
IF( NW.EQ.1 ) THEN
*
* Real 1x1 system.
*
* C = ca A - w D
*
CSR = CA*A( 1, 1 ) - WR*D1
CNORM = ABS( CSR )
*
* If | C | < SMINI, use C = SMINI
*
IF( CNORM.LT.SMINI ) THEN
CSR = SMINI
CNORM = SMINI
INFO = 1
END IF
*
* Check scaling for X = B / C
*
BNORM = ABS( B( 1, 1 ) )
IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN
IF( BNORM.GT.BIGNUM*CNORM )
$ SCALE = ONE / BNORM
END IF
*
* Compute X
*
X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / CSR
XNORM = ABS( X( 1, 1 ) )
ELSE
*
* Complex 1x1 system (w is complex)
*
* C = ca A - w D
*
CSR = CA*A( 1, 1 ) - WR*D1
CSI = -WI*D1
CNORM = ABS( CSR ) + ABS( CSI )
*
* If | C | < SMINI, use C = SMINI
*
IF( CNORM.LT.SMINI ) THEN
CSR = SMINI
CSI = ZERO
CNORM = SMINI
INFO = 1
END IF
*
* Check scaling for X = B / C
*
BNORM = ABS( B( 1, 1 ) ) + ABS( B( 1, 2 ) )
IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN
IF( BNORM.GT.BIGNUM*CNORM )
$ SCALE = ONE / BNORM
END IF
*
* Compute X
*
CALL DLADIV( SCALE*B( 1, 1 ), SCALE*B( 1, 2 ), CSR, CSI,
$ X( 1, 1 ), X( 1, 2 ) )
XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
END IF
*
ELSE
*
* 2x2 System
*
* Compute the real part of C = ca A - w D (or ca A**T - w D )
*
CR( 1, 1 ) = CA*A( 1, 1 ) - WR*D1
CR( 2, 2 ) = CA*A( 2, 2 ) - WR*D2
IF( LTRANS ) THEN
CR( 1, 2 ) = CA*A( 2, 1 )
CR( 2, 1 ) = CA*A( 1, 2 )
ELSE
CR( 2, 1 ) = CA*A( 2, 1 )
CR( 1, 2 ) = CA*A( 1, 2 )
END IF
*
IF( NW.EQ.1 ) THEN
*
* Real 2x2 system (w is real)
*
* Find the largest element in C
*
CMAX = ZERO
ICMAX = 0
*
DO 10 J = 1, 4
IF( ABS( CRV( J ) ).GT.CMAX ) THEN
CMAX = ABS( CRV( J ) )
ICMAX = J
END IF
10 CONTINUE
*
* If norm(C) < SMINI, use SMINI*identity.
*
IF( CMAX.LT.SMINI ) THEN
BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 2, 1 ) ) )
IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN
IF( BNORM.GT.BIGNUM*SMINI )
$ SCALE = ONE / BNORM
END IF
TEMP = SCALE / SMINI
X( 1, 1 ) = TEMP*B( 1, 1 )
X( 2, 1 ) = TEMP*B( 2, 1 )
XNORM = TEMP*BNORM
INFO = 1
RETURN
END IF
*
* Gaussian elimination with complete pivoting.
*
UR11 = CRV( ICMAX )
CR21 = CRV( IPIVOT( 2, ICMAX ) )
UR12 = CRV( IPIVOT( 3, ICMAX ) )
CR22 = CRV( IPIVOT( 4, ICMAX ) )
UR11R = ONE / UR11
LR21 = UR11R*CR21
UR22 = CR22 - UR12*LR21
*
* If smaller pivot < SMINI, use SMINI
*
IF( ABS( UR22 ).LT.SMINI ) THEN
UR22 = SMINI
INFO = 1
END IF
IF( RSWAP( ICMAX ) ) THEN
BR1 = B( 2, 1 )
BR2 = B( 1, 1 )
ELSE
BR1 = B( 1, 1 )
BR2 = B( 2, 1 )
END IF
BR2 = BR2 - LR21*BR1
BBND = MAX( ABS( BR1*( UR22*UR11R ) ), ABS( BR2 ) )
IF( BBND.GT.ONE .AND. ABS( UR22 ).LT.ONE ) THEN
IF( BBND.GE.BIGNUM*ABS( UR22 ) )
$ SCALE = ONE / BBND
END IF
*
XR2 = ( BR2*SCALE ) / UR22
XR1 = ( SCALE*BR1 )*UR11R - XR2*( UR11R*UR12 )
IF( ZSWAP( ICMAX ) ) THEN
X( 1, 1 ) = XR2
X( 2, 1 ) = XR1
ELSE
X( 1, 1 ) = XR1
X( 2, 1 ) = XR2
END IF
XNORM = MAX( ABS( XR1 ), ABS( XR2 ) )
*
* Further scaling if norm(A) norm(X) > overflow
*
IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN
IF( XNORM.GT.BIGNUM / CMAX ) THEN
TEMP = CMAX / BIGNUM
X( 1, 1 ) = TEMP*X( 1, 1 )
X( 2, 1 ) = TEMP*X( 2, 1 )
XNORM = TEMP*XNORM
SCALE = TEMP*SCALE
END IF
END IF
ELSE
*
* Complex 2x2 system (w is complex)
*
* Find the largest element in C
*
CI( 1, 1 ) = -WI*D1
CI( 2, 1 ) = ZERO
CI( 1, 2 ) = ZERO
CI( 2, 2 ) = -WI*D2
CMAX = ZERO
ICMAX = 0
*
DO 20 J = 1, 4
IF( ABS( CRV( J ) )+ABS( CIV( J ) ).GT.CMAX ) THEN
CMAX = ABS( CRV( J ) ) + ABS( CIV( J ) )
ICMAX = J
END IF
20 CONTINUE
*
* If norm(C) < SMINI, use SMINI*identity.
*
IF( CMAX.LT.SMINI ) THEN
BNORM = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ),
$ ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) )
IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN
IF( BNORM.GT.BIGNUM*SMINI )
$ SCALE = ONE / BNORM
END IF
TEMP = SCALE / SMINI
X( 1, 1 ) = TEMP*B( 1, 1 )
X( 2, 1 ) = TEMP*B( 2, 1 )
X( 1, 2 ) = TEMP*B( 1, 2 )
X( 2, 2 ) = TEMP*B( 2, 2 )
XNORM = TEMP*BNORM
INFO = 1
RETURN
END IF
*
* Gaussian elimination with complete pivoting.
*
UR11 = CRV( ICMAX )
UI11 = CIV( ICMAX )
CR21 = CRV( IPIVOT( 2, ICMAX ) )
CI21 = CIV( IPIVOT( 2, ICMAX ) )
UR12 = CRV( IPIVOT( 3, ICMAX ) )
UI12 = CIV( IPIVOT( 3, ICMAX ) )
CR22 = CRV( IPIVOT( 4, ICMAX ) )
CI22 = CIV( IPIVOT( 4, ICMAX ) )
IF( ICMAX.EQ.1 .OR. ICMAX.EQ.4 ) THEN
*
* Code when off-diagonals of pivoted C are real
*
IF( ABS( UR11 ).GT.ABS( UI11 ) ) THEN
TEMP = UI11 / UR11
UR11R = ONE / ( UR11*( ONE+TEMP**2 ) )
UI11R = -TEMP*UR11R
ELSE
TEMP = UR11 / UI11
UI11R = -ONE / ( UI11*( ONE+TEMP**2 ) )
UR11R = -TEMP*UI11R
END IF
LR21 = CR21*UR11R
LI21 = CR21*UI11R
UR12S = UR12*UR11R
UI12S = UR12*UI11R
UR22 = CR22 - UR12*LR21
UI22 = CI22 - UR12*LI21
ELSE
*
* Code when diagonals of pivoted C are real
*
UR11R = ONE / UR11
UI11R = ZERO
LR21 = CR21*UR11R
LI21 = CI21*UR11R
UR12S = UR12*UR11R
UI12S = UI12*UR11R
UR22 = CR22 - UR12*LR21 + UI12*LI21
UI22 = -UR12*LI21 - UI12*LR21
END IF
U22ABS = ABS( UR22 ) + ABS( UI22 )
*
* If smaller pivot < SMINI, use SMINI
*
IF( U22ABS.LT.SMINI ) THEN
UR22 = SMINI
UI22 = ZERO
INFO = 1
END IF
IF( RSWAP( ICMAX ) ) THEN
BR2 = B( 1, 1 )
BR1 = B( 2, 1 )
BI2 = B( 1, 2 )
BI1 = B( 2, 2 )
ELSE
BR1 = B( 1, 1 )
BR2 = B( 2, 1 )
BI1 = B( 1, 2 )
BI2 = B( 2, 2 )
END IF
BR2 = BR2 - LR21*BR1 + LI21*BI1
BI2 = BI2 - LI21*BR1 - LR21*BI1
BBND = MAX( ( ABS( BR1 )+ABS( BI1 ) )*
$ ( U22ABS*( ABS( UR11R )+ABS( UI11R ) ) ),
$ ABS( BR2 )+ABS( BI2 ) )
IF( BBND.GT.ONE .AND. U22ABS.LT.ONE ) THEN
IF( BBND.GE.BIGNUM*U22ABS ) THEN
SCALE = ONE / BBND
BR1 = SCALE*BR1
BI1 = SCALE*BI1
BR2 = SCALE*BR2
BI2 = SCALE*BI2
END IF
END IF
*
CALL DLADIV( BR2, BI2, UR22, UI22, XR2, XI2 )
XR1 = UR11R*BR1 - UI11R*BI1 - UR12S*XR2 + UI12S*XI2
XI1 = UI11R*BR1 + UR11R*BI1 - UI12S*XR2 - UR12S*XI2
IF( ZSWAP( ICMAX ) ) THEN
X( 1, 1 ) = XR2
X( 2, 1 ) = XR1
X( 1, 2 ) = XI2
X( 2, 2 ) = XI1
ELSE
X( 1, 1 ) = XR1
X( 2, 1 ) = XR2
X( 1, 2 ) = XI1
X( 2, 2 ) = XI2
END IF
XNORM = MAX( ABS( XR1 )+ABS( XI1 ), ABS( XR2 )+ABS( XI2 ) )
*
* Further scaling if norm(A) norm(X) > overflow
*
IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN
IF( XNORM.GT.BIGNUM / CMAX ) THEN
TEMP = CMAX / BIGNUM
X( 1, 1 ) = TEMP*X( 1, 1 )
X( 2, 1 ) = TEMP*X( 2, 1 )
X( 1, 2 ) = TEMP*X( 1, 2 )
X( 2, 2 ) = TEMP*X( 2, 2 )
XNORM = TEMP*XNORM
SCALE = TEMP*SCALE
END IF
END IF
END IF
END IF
*
RETURN
*
* End of DLALN2
*
END
*> \brief \b DLAMCH
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLAMCH( CMACH )
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAMCH determines double precision machine parameters.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] CMACH
*> \verbatim
*> Specifies the value to be returned by DLAMCH:
*> = 'E' or 'e', DLAMCH := eps
*> = 'S' or 's , DLAMCH := sfmin
*> = 'B' or 'b', DLAMCH := base
*> = 'P' or 'p', DLAMCH := eps*base
*> = 'N' or 'n', DLAMCH := t
*> = 'R' or 'r', DLAMCH := rnd
*> = 'M' or 'm', DLAMCH := emin
*> = 'U' or 'u', DLAMCH := rmin
*> = 'L' or 'l', DLAMCH := emax
*> = 'O' or 'o', DLAMCH := rmax
*> where
*> eps = relative machine precision
*> sfmin = safe minimum, such that 1/sfmin does not overflow
*> base = base of the machine
*> prec = eps*base
*> t = number of (base) digits in the mantissa
*> rnd = 1.0 when rounding occurs in addition, 0.0 otherwise
*> emin = minimum exponent before (gradual) underflow
*> rmin = underflow threshold - base**(emin-1)
*> emax = largest exponent before overflow
*> rmax = overflow threshold - (base**emax)*(1-eps)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2015
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLAMCH( CMACH )
*
* -- LAPACK auxiliary routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2015
*
* .. Scalar Arguments ..
CHARACTER CMACH
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION RND, EPS, SFMIN, SMALL, RMACH
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC DIGITS, EPSILON, HUGE, MAXEXPONENT,
$ MINEXPONENT, RADIX, TINY
* ..
* .. Executable Statements ..
*
*
* Assume rounding, not chopping. Always.
*
RND = ONE
*
IF( ONE.EQ.RND ) THEN
EPS = EPSILON(ZERO) * 0.5
ELSE
EPS = EPSILON(ZERO)
END IF
*
IF( LSAME( CMACH, 'E' ) ) THEN
RMACH = EPS
ELSE IF( LSAME( CMACH, 'S' ) ) THEN
SFMIN = TINY(ZERO)
SMALL = ONE / HUGE(ZERO)
IF( SMALL.GE.SFMIN ) THEN
*
* Use SMALL plus a bit, to avoid the possibility of rounding
* causing overflow when computing 1/sfmin.
*
SFMIN = SMALL*( ONE+EPS )
END IF
RMACH = SFMIN
ELSE IF( LSAME( CMACH, 'B' ) ) THEN
RMACH = RADIX(ZERO)
ELSE IF( LSAME( CMACH, 'P' ) ) THEN
RMACH = EPS * RADIX(ZERO)
ELSE IF( LSAME( CMACH, 'N' ) ) THEN
RMACH = DIGITS(ZERO)
ELSE IF( LSAME( CMACH, 'R' ) ) THEN
RMACH = RND
ELSE IF( LSAME( CMACH, 'M' ) ) THEN
RMACH = MINEXPONENT(ZERO)
ELSE IF( LSAME( CMACH, 'U' ) ) THEN
RMACH = tiny(zero)
ELSE IF( LSAME( CMACH, 'L' ) ) THEN
RMACH = MAXEXPONENT(ZERO)
ELSE IF( LSAME( CMACH, 'O' ) ) THEN
RMACH = HUGE(ZERO)
ELSE
RMACH = ZERO
END IF
*
DLAMCH = RMACH
RETURN
*
* End of DLAMCH
*
END
************************************************************************
*> \brief \b DLAMC3
*> \details
*> \b Purpose:
*> \verbatim
*> DLAMC3 is intended to force A and B to be stored prior to doing
*> the addition of A and B , for use in situations where optimizers
*> might hold one of these in a register.
*> \endverbatim
*> \author LAPACK is a software package provided by Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..
*> \date November 2015
*> \ingroup auxOTHERauxiliary
*>
*> \param[in] A
*> \verbatim
*> A is a DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is a DOUBLE PRECISION
*> The values A and B.
*> \endverbatim
*>
DOUBLE PRECISION FUNCTION DLAMC3( A, B )
*
* -- LAPACK auxiliary routine (version 3.6.0) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2010
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B
* ..
* =====================================================================
*
* .. Executable Statements ..
*
DLAMC3 = A + B
*
RETURN
*
* End of DLAMC3
*
END
*
************************************************************************
*> \brief \b DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general rectangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANGE + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANGE( NORM, M, N, A, LDA, WORK )
*
* .. Scalar Arguments ..
* CHARACTER NORM
* INTEGER LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANGE returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of a
*> real matrix A.
*> \endverbatim
*>
*> \return DLANGE
*> \verbatim
*>
*> DLANGE = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANGE as described
*> above.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0. When M = 0,
*> DLANGE is set to zero.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0. When N = 0,
*> DLANGE is set to zero.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The m by n matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(M,1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*> where LWORK >= M when NORM = 'I'; otherwise, WORK is not
*> referenced.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleGEauxiliary
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANGE( NORM, M, N, A, LDA, WORK )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION SCALE, SUM, VALUE, TEMP
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MIN, SQRT
* ..
* .. Executable Statements ..
*
IF( MIN( M, N ).EQ.0 ) THEN
VALUE = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
VALUE = ZERO
DO 20 J = 1, N
DO 10 I = 1, M
TEMP = ABS( A( I, J ) )
IF( VALUE.LT.TEMP .OR. DISNAN( TEMP ) ) VALUE = TEMP
10 CONTINUE
20 CONTINUE
ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
*
* Find norm1(A).
*
VALUE = ZERO
DO 40 J = 1, N
SUM = ZERO
DO 30 I = 1, M
SUM = SUM + ABS( A( I, J ) )
30 CONTINUE
IF( VALUE.LT.SUM .OR. DISNAN( SUM ) ) VALUE = SUM
40 CONTINUE
ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
* Find normI(A).
*
DO 50 I = 1, M
WORK( I ) = ZERO
50 CONTINUE
DO 70 J = 1, N
DO 60 I = 1, M
WORK( I ) = WORK( I ) + ABS( A( I, J ) )
60 CONTINUE
70 CONTINUE
VALUE = ZERO
DO 80 I = 1, M
TEMP = WORK( I )
IF( VALUE.LT.TEMP .OR. DISNAN( TEMP ) ) VALUE = TEMP
80 CONTINUE
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
SCALE = ZERO
SUM = ONE
DO 90 J = 1, N
CALL DLASSQ( M, A( 1, J ), 1, SCALE, SUM )
90 CONTINUE
VALUE = SCALE*SQRT( SUM )
END IF
*
DLANGE = VALUE
RETURN
*
* End of DLANGE
*
END
*> \brief \b DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANV2 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
*> matrix in standard form:
*>
*> [ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ]
*> [ C D ] [ SN CS ] [ CC DD ] [-SN CS ]
*>
*> where either
*> 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
*> 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
*> conjugate eigenvalues.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION
*> On entry, the elements of the input matrix.
*> On exit, they are overwritten by the elements of the
*> standardised Schur form.
*> \endverbatim
*>
*> \param[out] RT1R
*> \verbatim
*> RT1R is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] RT1I
*> \verbatim
*> RT1I is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] RT2R
*> \verbatim
*> RT2R is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] RT2I
*> \verbatim
*> RT2I is DOUBLE PRECISION
*> The real and imaginary parts of the eigenvalues. If the
*> eigenvalues are a complex conjugate pair, RT1I > 0.
*> \endverbatim
*>
*> \param[out] CS
*> \verbatim
*> CS is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SN
*> \verbatim
*> SN is DOUBLE PRECISION
*> Parameters of the rotation matrix.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Modified by V. Sima, Research Institute for Informatics, Bucharest,
*> Romania, to reduce the risk of cancellation errors,
*> when computing real eigenvalues, and to ensure, if possible, that
*> abs(RT1R) >= abs(RT2R).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
DOUBLE PRECISION MULTPL
PARAMETER ( MULTPL = 4.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, BB, BCMAX, BCMIS, CC, CS1, DD, EPS, P, SAB,
$ SAC, SCALE, SIGMA, SN1, TAU, TEMP, Z
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLAPY2
EXTERNAL DLAMCH, DLAPY2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SIGN, SQRT
* ..
* .. Executable Statements ..
*
EPS = DLAMCH( 'P' )
IF( C.EQ.ZERO ) THEN
CS = ONE
SN = ZERO
GO TO 10
*
ELSE IF( B.EQ.ZERO ) THEN
*
* Swap rows and columns
*
CS = ZERO
SN = ONE
TEMP = D
D = A
A = TEMP
B = -C
C = ZERO
GO TO 10
ELSE IF( ( A-D ).EQ.ZERO .AND. SIGN( ONE, B ).NE.SIGN( ONE, C ) )
$ THEN
CS = ONE
SN = ZERO
GO TO 10
ELSE
*
TEMP = A - D
P = HALF*TEMP
BCMAX = MAX( ABS( B ), ABS( C ) )
BCMIS = MIN( ABS( B ), ABS( C ) )*SIGN( ONE, B )*SIGN( ONE, C )
SCALE = MAX( ABS( P ), BCMAX )
Z = ( P / SCALE )*P + ( BCMAX / SCALE )*BCMIS
*
* If Z is of the order of the machine accuracy, postpone the
* decision on the nature of eigenvalues
*
IF( Z.GE.MULTPL*EPS ) THEN
*
* Real eigenvalues. Compute A and D.
*
Z = P + SIGN( SQRT( SCALE )*SQRT( Z ), P )
A = D + Z
D = D - ( BCMAX / Z )*BCMIS
*
* Compute B and the rotation matrix
*
TAU = DLAPY2( C, Z )
CS = Z / TAU
SN = C / TAU
B = B - C
C = ZERO
ELSE
*
* Complex eigenvalues, or real (almost) equal eigenvalues.
* Make diagonal elements equal.
*
SIGMA = B + C
TAU = DLAPY2( SIGMA, TEMP )
CS = SQRT( HALF*( ONE+ABS( SIGMA ) / TAU ) )
SN = -( P / ( TAU*CS ) )*SIGN( ONE, SIGMA )
*
* Compute [ AA BB ] = [ A B ] [ CS -SN ]
* [ CC DD ] [ C D ] [ SN CS ]
*
AA = A*CS + B*SN
BB = -A*SN + B*CS
CC = C*CS + D*SN
DD = -C*SN + D*CS
*
* Compute [ A B ] = [ CS SN ] [ AA BB ]
* [ C D ] [-SN CS ] [ CC DD ]
*
A = AA*CS + CC*SN
B = BB*CS + DD*SN
C = -AA*SN + CC*CS
D = -BB*SN + DD*CS
*
TEMP = HALF*( A+D )
A = TEMP
D = TEMP
*
IF( C.NE.ZERO ) THEN
IF( B.NE.ZERO ) THEN
IF( SIGN( ONE, B ).EQ.SIGN( ONE, C ) ) THEN
*
* Real eigenvalues: reduce to upper triangular form
*
SAB = SQRT( ABS( B ) )
SAC = SQRT( ABS( C ) )
P = SIGN( SAB*SAC, C )
TAU = ONE / SQRT( ABS( B+C ) )
A = TEMP + P
D = TEMP - P
B = B - C
C = ZERO
CS1 = SAB*TAU
SN1 = SAC*TAU
TEMP = CS*CS1 - SN*SN1
SN = CS*SN1 + SN*CS1
CS = TEMP
END IF
ELSE
B = -C
C = ZERO
TEMP = CS
CS = -SN
SN = TEMP
END IF
END IF
END IF
*
END IF
*
10 CONTINUE
*
* Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I).
*
RT1R = A
RT2R = D
IF( C.EQ.ZERO ) THEN
RT1I = ZERO
RT2I = ZERO
ELSE
RT1I = SQRT( ABS( B ) )*SQRT( ABS( C ) )
RT2I = -RT1I
END IF
RETURN
*
* End of DLANV2
*
END
*> \brief \b DLAPY2 returns sqrt(x2+y2).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAPY2 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLAPY2( X, Y )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION X, Y
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary
*> overflow.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] Y
*> \verbatim
*> Y is DOUBLE PRECISION
*> X and Y specify the values x and y.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLAPY2( X, Y )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
DOUBLE PRECISION X, Y
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION W, XABS, YABS, Z
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
XABS = ABS( X )
YABS = ABS( Y )
W = MAX( XABS, YABS )
Z = MIN( XABS, YABS )
IF( Z.EQ.ZERO ) THEN
DLAPY2 = W
ELSE
DLAPY2 = W*SQRT( ONE+( Z / W )**2 )
END IF
RETURN
*
* End of DLAPY2
*
END
*> \brief \b DLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR0 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
* ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQR0 computes the eigenvalues of a Hessenberg matrix H
*> and, optionally, the matrices T and Z from the Schur decomposition
*> H = Z T Z**T, where T is an upper quasi-triangular matrix (the
*> Schur form), and Z is the orthogonal matrix of Schur vectors.
*>
*> Optionally Z may be postmultiplied into an input orthogonal
*> matrix Q so that this routine can give the Schur factorization
*> of a matrix A which has been reduced to the Hessenberg form H
*> by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> = .TRUE. : the full Schur form T is required;
*> = .FALSE.: only eigenvalues are required.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> = .TRUE. : the matrix of Schur vectors Z is required;
*> = .FALSE.: Schur vectors are not required.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N .GE. 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> It is assumed that H is already upper triangular in rows
*> and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
*> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
*> previous call to DGEBAL, and then passed to DGEHRD when the
*> matrix output by DGEBAL is reduced to Hessenberg form.
*> Otherwise, ILO and IHI should be set to 1 and N,
*> respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
*> If N = 0, then ILO = 1 and IHI = 0.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On entry, the upper Hessenberg matrix H.
*> On exit, if INFO = 0 and WANTT is .TRUE., then H contains
*> the upper quasi-triangular matrix T from the Schur
*> decomposition (the Schur form); 2-by-2 diagonal blocks
*> (corresponding to complex conjugate pairs of eigenvalues)
*> are returned in standard form, with H(i,i) = H(i+1,i+1)
*> and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
*> .FALSE., then the contents of H are unspecified on exit.
*> (The output value of H when INFO.GT.0 is given under the
*> description of INFO below.)
*>
*> This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
*> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH .GE. max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (IHI)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (IHI)
*> The real and imaginary parts, respectively, of the computed
*> eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
*> and WI(ILO:IHI). If two eigenvalues are computed as a
*> complex conjugate pair, they are stored in consecutive
*> elements of WR and WI, say the i-th and (i+1)th, with
*> WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
*> the eigenvalues are stored in the same order as on the
*> diagonal of the Schur form returned in H, with
*> WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
*> block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
*> WI(i+1) = -WI(i).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE..
*> 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,IHI)
*> If WANTZ is .FALSE., then Z is not referenced.
*> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
*> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
*> orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
*> (The output value of Z when INFO.GT.0 is given under
*> the description of INFO below.)
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. if WANTZ is .TRUE.
*> then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension LWORK
*> On exit, if LWORK = -1, WORK(1) returns an estimate of
*> the optimal value for LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK .GE. max(1,N)
*> is sufficient, but LWORK typically as large as 6*N may
*> be required for optimal performance. A workspace query
*> to determine the optimal workspace size is recommended.
*>
*> If LWORK = -1, then DLAQR0 does a workspace query.
*> In this case, DLAQR0 checks the input parameters and
*> estimates the optimal workspace size for the given
*> values of N, ILO and IHI. The estimate is returned
*> in WORK(1). No error message related to LWORK is
*> issued by XERBLA. Neither H nor Z are accessed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> .GT. 0: if INFO = i, DLAQR0 failed to compute all of
*> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
*> and WI contain those eigenvalues which have been
*> successfully computed. (Failures are rare.)
*>
*> If INFO .GT. 0 and WANT is .FALSE., then on exit,
*> the remaining unconverged eigenvalues are the eigen-
*> values of the upper Hessenberg matrix rows and
*> columns ILO through INFO of the final, output
*> value of H.
*>
*> If INFO .GT. 0 and WANTT is .TRUE., then on exit
*>
*> (*) (initial value of H)*U = U*(final value of H)
*>
*> where U is an orthogonal matrix. The final
*> value of H is upper Hessenberg and quasi-triangular
*> in rows and columns INFO+1 through IHI.
*>
*> If INFO .GT. 0 and WANTZ is .TRUE., then on exit
*>
*> (final value of Z(ILO:IHI,ILOZ:IHIZ)
*> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
*>
*> where U is the orthogonal matrix in (*) (regard-
*> less of the value of WANTT.)
*>
*> If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
*> accessed.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*
*> \par References:
* ================
*>
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
*> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
*> 929--947, 2002.
*> \n
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part II: Aggressive Early Deflation, SIAM Journal
*> of Matrix Analysis, volume 23, pages 948--973, 2002.
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
$ Z( LDZ, * )
* ..
*
* ================================================================
*
* .. Parameters ..
*
* ==== Matrices of order NTINY or smaller must be processed by
* . DLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
*
* ==== Exceptional deflation windows: try to cure rare
* . slow convergence by varying the size of the
* . deflation window after KEXNW iterations. ====
INTEGER KEXNW
PARAMETER ( KEXNW = 5 )
*
* ==== Exceptional shifts: try to cure rare slow convergence
* . with ad-hoc exceptional shifts every KEXSH iterations.
* . ====
INTEGER KEXSH
PARAMETER ( KEXSH = 6 )
*
* ==== The constants WILK1 and WILK2 are used to form the
* . exceptional shifts. ====
DOUBLE PRECISION WILK1, WILK2
PARAMETER ( WILK1 = 0.75d0, WILK2 = -0.4375d0 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP
INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
$ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
$ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
$ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
LOGICAL SORTED
CHARACTER JBCMPZ*2
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Local Arrays ..
DOUBLE PRECISION ZDUM( 1, 1 )
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DLAHQR, DLANV2, DLAQR3, DLAQR4, DLAQR5
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, MAX, MIN, MOD
* ..
* .. Executable Statements ..
INFO = 0
*
* ==== Quick return for N = 0: nothing to do. ====
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = ONE
RETURN
END IF
*
IF( N.LE.NTINY ) THEN
*
* ==== Tiny matrices must use DLAHQR. ====
*
LWKOPT = 1
IF( LWORK.NE.-1 )
$ CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, INFO )
ELSE
*
* ==== Use small bulge multi-shift QR with aggressive early
* . deflation on larger-than-tiny matrices. ====
*
* ==== Hope for the best. ====
*
INFO = 0
*
* ==== Set up job flags for ILAENV. ====
*
IF( WANTT ) THEN
JBCMPZ( 1: 1 ) = 'S'
ELSE
JBCMPZ( 1: 1 ) = 'E'
END IF
IF( WANTZ ) THEN
JBCMPZ( 2: 2 ) = 'V'
ELSE
JBCMPZ( 2: 2 ) = 'N'
END IF
*
* ==== NWR = recommended deflation window size. At this
* . point, N .GT. NTINY = 11, so there is enough
* . subdiagonal workspace for NWR.GE.2 as required.
* . (In fact, there is enough subdiagonal space for
* . NWR.GE.3.) ====
*
NWR = ILAENV( 13, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
* ==== NSR = recommended number of simultaneous shifts.
* . At this point N .GT. NTINY = 11, so there is at
* . enough subdiagonal workspace for NSR to be even
* . and greater than or equal to two as required. ====
*
NSR = ILAENV( 15, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
* ==== Estimate optimal workspace ====
*
* ==== Workspace query call to DLAQR3 ====
*
CALL DLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
$ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
$ N, H, LDH, WORK, -1 )
*
* ==== Optimal workspace = MAX(DLAQR5, DLAQR3) ====
*
LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
*
* ==== Quick return in case of workspace query. ====
*
IF( LWORK.EQ.-1 ) THEN
WORK( 1 ) = DBLE( LWKOPT )
RETURN
END IF
*
* ==== DLAHQR/DLAQR0 crossover point ====
*
NMIN = ILAENV( 12, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NMIN = MAX( NTINY, NMIN )
*
* ==== Nibble crossover point ====
*
NIBBLE = ILAENV( 14, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NIBBLE = MAX( 0, NIBBLE )
*
* ==== Accumulate reflections during ttswp? Use block
* . 2-by-2 structure during matrix-matrix multiply? ====
*
KACC22 = ILAENV( 16, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
KACC22 = MAX( 0, KACC22 )
KACC22 = MIN( 2, KACC22 )
*
* ==== NWMAX = the largest possible deflation window for
* . which there is sufficient workspace. ====
*
NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
NW = NWMAX
*
* ==== NSMAX = the Largest number of simultaneous shifts
* . for which there is sufficient workspace. ====
*
NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
NSMAX = NSMAX - MOD( NSMAX, 2 )
*
* ==== NDFL: an iteration count restarted at deflation. ====
*
NDFL = 1
*
* ==== ITMAX = iteration limit ====
*
ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
*
* ==== Last row and column in the active block ====
*
KBOT = IHI
*
* ==== Main Loop ====
*
DO 80 IT = 1, ITMAX
*
* ==== Done when KBOT falls below ILO ====
*
IF( KBOT.LT.ILO )
$ GO TO 90
*
* ==== Locate active block ====
*
DO 10 K = KBOT, ILO + 1, -1
IF( H( K, K-1 ).EQ.ZERO )
$ GO TO 20
10 CONTINUE
K = ILO
20 CONTINUE
KTOP = K
*
* ==== Select deflation window size:
* . Typical Case:
* . If possible and advisable, nibble the entire
* . active block. If not, use size MIN(NWR,NWMAX)
* . or MIN(NWR+1,NWMAX) depending upon which has
* . the smaller corresponding subdiagonal entry
* . (a heuristic).
* .
* . Exceptional Case:
* . If there have been no deflations in KEXNW or
* . more iterations, then vary the deflation window
* . size. At first, because, larger windows are,
* . in general, more powerful than smaller ones,
* . rapidly increase the window to the maximum possible.
* . Then, gradually reduce the window size. ====
*
NH = KBOT - KTOP + 1
NWUPBD = MIN( NH, NWMAX )
IF( NDFL.LT.KEXNW ) THEN
NW = MIN( NWUPBD, NWR )
ELSE
NW = MIN( NWUPBD, 2*NW )
END IF
IF( NW.LT.NWMAX ) THEN
IF( NW.GE.NH-1 ) THEN
NW = NH
ELSE
KWTOP = KBOT - NW + 1
IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
$ ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
END IF
END IF
IF( NDFL.LT.KEXNW ) THEN
NDEC = -1
ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
NDEC = NDEC + 1
IF( NW-NDEC.LT.2 )
$ NDEC = 0
NW = NW - NDEC
END IF
*
* ==== Aggressive early deflation:
* . split workspace under the subdiagonal into
* . - an nw-by-nw work array V in the lower
* . left-hand-corner,
* . - an NW-by-at-least-NW-but-more-is-better
* . (NW-by-NHO) horizontal work array along
* . the bottom edge,
* . - an at-least-NW-but-more-is-better (NHV-by-NW)
* . vertical work array along the left-hand-edge.
* . ====
*
KV = N - NW + 1
KT = NW + 1
NHO = ( N-NW-1 ) - KT + 1
KWV = NW + 2
NVE = ( N-NW ) - KWV + 1
*
* ==== Aggressive early deflation ====
*
CALL DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
$ IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
$ NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
$ WORK, LWORK )
*
* ==== Adjust KBOT accounting for new deflations. ====
*
KBOT = KBOT - LD
*
* ==== KS points to the shifts. ====
*
KS = KBOT - LS + 1
*
* ==== Skip an expensive QR sweep if there is a (partly
* . heuristic) reason to expect that many eigenvalues
* . will deflate without it. Here, the QR sweep is
* . skipped if many eigenvalues have just been deflated
* . or if the remaining active block is small.
*
IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
$ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
*
* ==== NS = nominal number of simultaneous shifts.
* . This may be lowered (slightly) if DLAQR3
* . did not provide that many shifts. ====
*
NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
NS = NS - MOD( NS, 2 )
*
* ==== If there have been no deflations
* . in a multiple of KEXSH iterations,
* . then try exceptional shifts.
* . Otherwise use shifts provided by
* . DLAQR3 above or from the eigenvalues
* . of a trailing principal submatrix. ====
*
IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
KS = KBOT - NS + 1
DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
AA = WILK1*SS + H( I, I )
BB = SS
CC = WILK2*SS
DD = AA
CALL DLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ),
$ WR( I ), WI( I ), CS, SN )
30 CONTINUE
IF( KS.EQ.KTOP ) THEN
WR( KS+1 ) = H( KS+1, KS+1 )
WI( KS+1 ) = ZERO
WR( KS ) = WR( KS+1 )
WI( KS ) = WI( KS+1 )
END IF
ELSE
*
* ==== Got NS/2 or fewer shifts? Use DLAQR4 or
* . DLAHQR on a trailing principal submatrix to
* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
* . there is enough space below the subdiagonal
* . to fit an NS-by-NS scratch array.) ====
*
IF( KBOT-KS+1.LE.NS / 2 ) THEN
KS = KBOT - NS + 1
KT = N - NS + 1
CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH,
$ H( KT, 1 ), LDH )
IF( NS.GT.NMIN ) THEN
CALL DLAQR4( .false., .false., NS, 1, NS,
$ H( KT, 1 ), LDH, WR( KS ),
$ WI( KS ), 1, 1, ZDUM, 1, WORK,
$ LWORK, INF )
ELSE
CALL DLAHQR( .false., .false., NS, 1, NS,
$ H( KT, 1 ), LDH, WR( KS ),
$ WI( KS ), 1, 1, ZDUM, 1, INF )
END IF
KS = KS + INF
*
* ==== In case of a rare QR failure use
* . eigenvalues of the trailing 2-by-2
* . principal submatrix. ====
*
IF( KS.GE.KBOT ) THEN
AA = H( KBOT-1, KBOT-1 )
CC = H( KBOT, KBOT-1 )
BB = H( KBOT-1, KBOT )
DD = H( KBOT, KBOT )
CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
$ WI( KBOT-1 ), WR( KBOT ),
$ WI( KBOT ), CS, SN )
KS = KBOT - 1
END IF
END IF
*
IF( KBOT-KS+1.GT.NS ) THEN
*
* ==== Sort the shifts (Helps a little)
* . Bubble sort keeps complex conjugate
* . pairs together. ====
*
SORTED = .false.
DO 50 K = KBOT, KS + 1, -1
IF( SORTED )
$ GO TO 60
SORTED = .true.
DO 40 I = KS, K - 1
IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
$ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
SORTED = .false.
*
SWAP = WR( I )
WR( I ) = WR( I+1 )
WR( I+1 ) = SWAP
*
SWAP = WI( I )
WI( I ) = WI( I+1 )
WI( I+1 ) = SWAP
END IF
40 CONTINUE
50 CONTINUE
60 CONTINUE
END IF
*
* ==== Shuffle shifts into pairs of real shifts
* . and pairs of complex conjugate shifts
* . assuming complex conjugate shifts are
* . already adjacent to one another. (Yes,
* . they are.) ====
*
DO 70 I = KBOT, KS + 2, -2
IF( WI( I ).NE.-WI( I-1 ) ) THEN
*
SWAP = WR( I )
WR( I ) = WR( I-1 )
WR( I-1 ) = WR( I-2 )
WR( I-2 ) = SWAP
*
SWAP = WI( I )
WI( I ) = WI( I-1 )
WI( I-1 ) = WI( I-2 )
WI( I-2 ) = SWAP
END IF
70 CONTINUE
END IF
*
* ==== If there are only two shifts and both are
* . real, then use only one. ====
*
IF( KBOT-KS+1.EQ.2 ) THEN
IF( WI( KBOT ).EQ.ZERO ) THEN
IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
$ ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
WR( KBOT-1 ) = WR( KBOT )
ELSE
WR( KBOT ) = WR( KBOT-1 )
END IF
END IF
END IF
*
* ==== Use up to NS of the the smallest magnatiude
* . shifts. If there aren't NS shifts available,
* . then use them all, possibly dropping one to
* . make the number of shifts even. ====
*
NS = MIN( NS, KBOT-KS+1 )
NS = NS - MOD( NS, 2 )
KS = KBOT - NS + 1
*
* ==== Small-bulge multi-shift QR sweep:
* . split workspace under the subdiagonal into
* . - a KDU-by-KDU work array U in the lower
* . left-hand-corner,
* . - a KDU-by-at-least-KDU-but-more-is-better
* . (KDU-by-NHo) horizontal work array WH along
* . the bottom edge,
* . - and an at-least-KDU-but-more-is-better-by-KDU
* . (NVE-by-KDU) vertical work WV arrow along
* . the left-hand-edge. ====
*
KDU = 3*NS - 3
KU = N - KDU + 1
KWH = KDU + 1
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
KWV = KDU + 4
NVE = N - KDU - KWV + 1
*
* ==== Small-bulge multi-shift QR sweep ====
*
CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
$ WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
$ LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
$ H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
END IF
*
* ==== Note progress (or the lack of it). ====
*
IF( LD.GT.0 ) THEN
NDFL = 1
ELSE
NDFL = NDFL + 1
END IF
*
* ==== End of main loop ====
80 CONTINUE
*
* ==== Iteration limit exceeded. Set INFO to show where
* . the problem occurred and exit. ====
*
INFO = KBOT
90 CONTINUE
END IF
*
* ==== Return the optimal value of LWORK. ====
*
WORK( 1 ) = DBLE( LWKOPT )
*
* ==== End of DLAQR0 ====
*
END
*> \brief \b DLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR1 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQR1( N, H, LDH, SR1, SI1, SR2, SI2, V )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION SI1, SI2, SR1, SR2
* INTEGER LDH, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), V( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Given a 2-by-2 or 3-by-3 matrix H, DLAQR1 sets v to a
*> scalar multiple of the first column of the product
*>
*> (*) K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)
*>
*> scaling to avoid overflows and most underflows. It
*> is assumed that either
*>
*> 1) sr1 = sr2 and si1 = -si2
*> or
*> 2) si1 = si2 = 0.
*>
*> This is useful for starting double implicit shift bulges
*> in the QR algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is integer
*> Order of the matrix H. N must be either 2 or 3.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*> H is DOUBLE PRECISION array of dimension (LDH,N)
*> The 2-by-2 or 3-by-3 matrix H in (*).
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is integer
*> The leading dimension of H as declared in
*> the calling procedure. LDH.GE.N
*> \endverbatim
*>
*> \param[in] SR1
*> \verbatim
*> SR1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] SI1
*> \verbatim
*> SI1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] SR2
*> \verbatim
*> SR2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] SI2
*> \verbatim
*> SI2 is DOUBLE PRECISION
*> The shifts in (*).
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array of dimension N
*> A scalar multiple of the first column of the
*> matrix K in (*).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*>
* =====================================================================
SUBROUTINE DLAQR1( N, H, LDH, SR1, SI1, SR2, SI2, V )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
DOUBLE PRECISION SI1, SI2, SR1, SR2
INTEGER LDH, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), V( * )
* ..
*
* ================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION H21S, H31S, S
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
IF( N.EQ.2 ) THEN
S = ABS( H( 1, 1 )-SR2 ) + ABS( SI2 ) + ABS( H( 2, 1 ) )
IF( S.EQ.ZERO ) THEN
V( 1 ) = ZERO
V( 2 ) = ZERO
ELSE
H21S = H( 2, 1 ) / S
V( 1 ) = H21S*H( 1, 2 ) + ( H( 1, 1 )-SR1 )*
$ ( ( H( 1, 1 )-SR2 ) / S ) - SI1*( SI2 / S )
V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-SR1-SR2 )
END IF
ELSE
S = ABS( H( 1, 1 )-SR2 ) + ABS( SI2 ) + ABS( H( 2, 1 ) ) +
$ ABS( H( 3, 1 ) )
IF( S.EQ.ZERO ) THEN
V( 1 ) = ZERO
V( 2 ) = ZERO
V( 3 ) = ZERO
ELSE
H21S = H( 2, 1 ) / S
H31S = H( 3, 1 ) / S
V( 1 ) = ( H( 1, 1 )-SR1 )*( ( H( 1, 1 )-SR2 ) / S ) -
$ SI1*( SI2 / S ) + H( 1, 2 )*H21S + H( 1, 3 )*H31S
V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-SR1-SR2 ) +
$ H( 2, 3 )*H31S
V( 3 ) = H31S*( H( 1, 1 )+H( 3, 3 )-SR1-SR2 ) +
$ H21S*H( 3, 2 )
END IF
END IF
END
*> \brief \b DLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR2 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
* IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
* LDT, NV, WV, LDWV, WORK, LWORK )
*
* .. Scalar Arguments ..
* INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
* $ LDZ, LWORK, N, ND, NH, NS, NV, NW
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
* $ V( LDV, * ), WORK( * ), WV( LDWV, * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQR2 is identical to DLAQR3 except that it avoids
*> recursion by calling DLAHQR instead of DLAQR4.
*>
*> Aggressive early deflation:
*>
*> This subroutine accepts as input an upper Hessenberg matrix
*> H and performs an orthogonal similarity transformation
*> designed to detect and deflate fully converged eigenvalues from
*> a trailing principal submatrix. On output H has been over-
*> written by a new Hessenberg matrix that is a perturbation of
*> an orthogonal similarity transformation of H. It is to be
*> hoped that the final version of H has many zero subdiagonal
*> entries.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> If .TRUE., then the Hessenberg matrix H is fully updated
*> so that the quasi-triangular Schur factor may be
*> computed (in cooperation with the calling subroutine).
*> If .FALSE., then only enough of H is updated to preserve
*> the eigenvalues.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> If .TRUE., then the orthogonal matrix Z is updated so
*> so that the orthogonal Schur factor may be computed
*> (in cooperation with the calling subroutine).
*> If .FALSE., then Z is not referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H and (if WANTZ is .TRUE.) the
*> order of the orthogonal matrix Z.
*> \endverbatim
*>
*> \param[in] KTOP
*> \verbatim
*> KTOP is INTEGER
*> It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
*> KBOT and KTOP together determine an isolated block
*> along the diagonal of the Hessenberg matrix.
*> \endverbatim
*>
*> \param[in] KBOT
*> \verbatim
*> KBOT is INTEGER
*> It is assumed without a check that either
*> KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
*> determine an isolated block along the diagonal of the
*> Hessenberg matrix.
*> \endverbatim
*>
*> \param[in] NW
*> \verbatim
*> NW is INTEGER
*> Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On input the initial N-by-N section of H stores the
*> Hessenberg matrix undergoing aggressive early deflation.
*> On output H has been transformed by an orthogonal
*> similarity transformation, perturbed, and the returned
*> to Hessenberg form that (it is to be hoped) has some
*> zero subdiagonal entries.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is integer
*> Leading dimension of H just as declared in the calling
*> subroutine. N .LE. LDH
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> IF WANTZ is .TRUE., then on output, the orthogonal
*> similarity transformation mentioned above has been
*> accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
*> If WANTZ is .FALSE., then Z is unreferenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is integer
*> The leading dimension of Z just as declared in the
*> calling subroutine. 1 .LE. LDZ.
*> \endverbatim
*>
*> \param[out] NS
*> \verbatim
*> NS is integer
*> The number of unconverged (ie approximate) eigenvalues
*> returned in SR and SI that may be used as shifts by the
*> calling subroutine.
*> \endverbatim
*>
*> \param[out] ND
*> \verbatim
*> ND is integer
*> The number of converged eigenvalues uncovered by this
*> subroutine.
*> \endverbatim
*>
*> \param[out] SR
*> \verbatim
*> SR is DOUBLE PRECISION array, dimension (KBOT)
*> \endverbatim
*>
*> \param[out] SI
*> \verbatim
*> SI is DOUBLE PRECISION array, dimension (KBOT)
*> On output, the real and imaginary parts of approximate
*> eigenvalues that may be used for shifts are stored in
*> SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
*> SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
*> The real and imaginary parts of converged eigenvalues
*> are stored in SR(KBOT-ND+1) through SR(KBOT) and
*> SI(KBOT-ND+1) through SI(KBOT), respectively.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,NW)
*> An NW-by-NW work array.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is integer scalar
*> The leading dimension of V just as declared in the
*> calling subroutine. NW .LE. LDV
*> \endverbatim
*>
*> \param[in] NH
*> \verbatim
*> NH is integer scalar
*> The number of columns of T. NH.GE.NW.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,NW)
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is integer
*> The leading dimension of T just as declared in the
*> calling subroutine. NW .LE. LDT
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*> NV is integer
*> The number of rows of work array WV available for
*> workspace. NV.GE.NW.
*> \endverbatim
*>
*> \param[out] WV
*> \verbatim
*> WV is DOUBLE PRECISION array, dimension (LDWV,NW)
*> \endverbatim
*>
*> \param[in] LDWV
*> \verbatim
*> LDWV is integer
*> The leading dimension of W just as declared in the
*> calling subroutine. NW .LE. LDV
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> On exit, WORK(1) is set to an estimate of the optimal value
*> of LWORK for the given values of N, NW, KTOP and KBOT.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is integer
*> The dimension of the work array WORK. LWORK = 2*NW
*> suffices, but greater efficiency may result from larger
*> values of LWORK.
*>
*> If LWORK = -1, then a workspace query is assumed; DLAQR2
*> only estimates the optimal workspace size for the given
*> values of N, NW, KTOP and KBOT. The estimate is returned
*> in WORK(1). No error message related to LWORK is issued
*> by XERBLA. Neither H nor Z are accessed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*>
* =====================================================================
SUBROUTINE DLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
$ IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
$ LDT, NV, WV, LDWV, WORK, LWORK )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
$ LDZ, LWORK, N, ND, NH, NS, NV, NW
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
$ V( LDV, * ), WORK( * ), WV( LDWV, * ),
$ Z( LDZ, * )
* ..
*
* ================================================================
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, BB, BETA, CC, CS, DD, EVI, EVK, FOO, S,
$ SAFMAX, SAFMIN, SMLNUM, SN, TAU, ULP
INTEGER I, IFST, ILST, INFO, INFQR, J, JW, K, KCOL,
$ KEND, KLN, KROW, KWTOP, LTOP, LWK1, LWK2,
$ LWKOPT
LOGICAL BULGE, SORTED
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEHRD, DGEMM, DLABAD, DLACPY, DLAHQR,
$ DLANV2, DLARF, DLARFG, DLASET, DORMHR, DTREXC
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* ==== Estimate optimal workspace. ====
*
JW = MIN( NW, KBOT-KTOP+1 )
IF( JW.LE.2 ) THEN
LWKOPT = 1
ELSE
*
* ==== Workspace query call to DGEHRD ====
*
CALL DGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
LWK1 = INT( WORK( 1 ) )
*
* ==== Workspace query call to DORMHR ====
*
CALL DORMHR( 'R', 'N', JW, JW, 1, JW-1, T, LDT, WORK, V, LDV,
$ WORK, -1, INFO )
LWK2 = INT( WORK( 1 ) )
*
* ==== Optimal workspace ====
*
LWKOPT = JW + MAX( LWK1, LWK2 )
END IF
*
* ==== Quick return in case of workspace query. ====
*
IF( LWORK.EQ.-1 ) THEN
WORK( 1 ) = DBLE( LWKOPT )
RETURN
END IF
*
* ==== Nothing to do ...
* ... for an empty active block ... ====
NS = 0
ND = 0
WORK( 1 ) = ONE
IF( KTOP.GT.KBOT )
$ RETURN
* ... nor for an empty deflation window. ====
IF( NW.LT.1 )
$ RETURN
*
* ==== Machine constants ====
*
SAFMIN = DLAMCH( 'SAFE MINIMUM' )
SAFMAX = ONE / SAFMIN
CALL DLABAD( SAFMIN, SAFMAX )
ULP = DLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( DBLE( N ) / ULP )
*
* ==== Setup deflation window ====
*
JW = MIN( NW, KBOT-KTOP+1 )
KWTOP = KBOT - JW + 1
IF( KWTOP.EQ.KTOP ) THEN
S = ZERO
ELSE
S = H( KWTOP, KWTOP-1 )
END IF
*
IF( KBOT.EQ.KWTOP ) THEN
*
* ==== 1-by-1 deflation window: not much to do ====
*
SR( KWTOP ) = H( KWTOP, KWTOP )
SI( KWTOP ) = ZERO
NS = 1
ND = 0
IF( ABS( S ).LE.MAX( SMLNUM, ULP*ABS( H( KWTOP, KWTOP ) ) ) )
$ THEN
NS = 0
ND = 1
IF( KWTOP.GT.KTOP )
$ H( KWTOP, KWTOP-1 ) = ZERO
END IF
WORK( 1 ) = ONE
RETURN
END IF
*
* ==== Convert to spike-triangular form. (In case of a
* . rare QR failure, this routine continues to do
* . aggressive early deflation using that part of
* . the deflation window that converged using INFQR
* . here and there to keep track.) ====
*
CALL DLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
CALL DCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ), LDT+1 )
*
CALL DLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
CALL DLAHQR( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
$ SI( KWTOP ), 1, JW, V, LDV, INFQR )
*
* ==== DTREXC needs a clean margin near the diagonal ====
*
DO 10 J = 1, JW - 3
T( J+2, J ) = ZERO
T( J+3, J ) = ZERO
10 CONTINUE
IF( JW.GT.2 )
$ T( JW, JW-2 ) = ZERO
*
* ==== Deflation detection loop ====
*
NS = JW
ILST = INFQR + 1
20 CONTINUE
IF( ILST.LE.NS ) THEN
IF( NS.EQ.1 ) THEN
BULGE = .FALSE.
ELSE
BULGE = T( NS, NS-1 ).NE.ZERO
END IF
*
* ==== Small spike tip test for deflation ====
*
IF( .NOT.BULGE ) THEN
*
* ==== Real eigenvalue ====
*
FOO = ABS( T( NS, NS ) )
IF( FOO.EQ.ZERO )
$ FOO = ABS( S )
IF( ABS( S*V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) ) THEN
*
* ==== Deflatable ====
*
NS = NS - 1
ELSE
*
* ==== Undeflatable. Move it up out of the way.
* . (DTREXC can not fail in this case.) ====
*
IFST = NS
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
$ INFO )
ILST = ILST + 1
END IF
ELSE
*
* ==== Complex conjugate pair ====
*
FOO = ABS( T( NS, NS ) ) + SQRT( ABS( T( NS, NS-1 ) ) )*
$ SQRT( ABS( T( NS-1, NS ) ) )
IF( FOO.EQ.ZERO )
$ FOO = ABS( S )
IF( MAX( ABS( S*V( 1, NS ) ), ABS( S*V( 1, NS-1 ) ) ).LE.
$ MAX( SMLNUM, ULP*FOO ) ) THEN
*
* ==== Deflatable ====
*
NS = NS - 2
ELSE
*
* ==== Undeflatable. Move them up out of the way.
* . Fortunately, DTREXC does the right thing with
* . ILST in case of a rare exchange failure. ====
*
IFST = NS
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
$ INFO )
ILST = ILST + 2
END IF
END IF
*
* ==== End deflation detection loop ====
*
GO TO 20
END IF
*
* ==== Return to Hessenberg form ====
*
IF( NS.EQ.0 )
$ S = ZERO
*
IF( NS.LT.JW ) THEN
*
* ==== sorting diagonal blocks of T improves accuracy for
* . graded matrices. Bubble sort deals well with
* . exchange failures. ====
*
SORTED = .false.
I = NS + 1
30 CONTINUE
IF( SORTED )
$ GO TO 50
SORTED = .true.
*
KEND = I - 1
I = INFQR + 1
IF( I.EQ.NS ) THEN
K = I + 1
ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
K = I + 1
ELSE
K = I + 2
END IF
40 CONTINUE
IF( K.LE.KEND ) THEN
IF( K.EQ.I+1 ) THEN
EVI = ABS( T( I, I ) )
ELSE
EVI = ABS( T( I, I ) ) + SQRT( ABS( T( I+1, I ) ) )*
$ SQRT( ABS( T( I, I+1 ) ) )
END IF
*
IF( K.EQ.KEND ) THEN
EVK = ABS( T( K, K ) )
ELSE IF( T( K+1, K ).EQ.ZERO ) THEN
EVK = ABS( T( K, K ) )
ELSE
EVK = ABS( T( K, K ) ) + SQRT( ABS( T( K+1, K ) ) )*
$ SQRT( ABS( T( K, K+1 ) ) )
END IF
*
IF( EVI.GE.EVK ) THEN
I = K
ELSE
SORTED = .false.
IFST = I
ILST = K
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
$ INFO )
IF( INFO.EQ.0 ) THEN
I = ILST
ELSE
I = K
END IF
END IF
IF( I.EQ.KEND ) THEN
K = I + 1
ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
K = I + 1
ELSE
K = I + 2
END IF
GO TO 40
END IF
GO TO 30
50 CONTINUE
END IF
*
* ==== Restore shift/eigenvalue array from T ====
*
I = JW
60 CONTINUE
IF( I.GE.INFQR+1 ) THEN
IF( I.EQ.INFQR+1 ) THEN
SR( KWTOP+I-1 ) = T( I, I )
SI( KWTOP+I-1 ) = ZERO
I = I - 1
ELSE IF( T( I, I-1 ).EQ.ZERO ) THEN
SR( KWTOP+I-1 ) = T( I, I )
SI( KWTOP+I-1 ) = ZERO
I = I - 1
ELSE
AA = T( I-1, I-1 )
CC = T( I, I-1 )
BB = T( I-1, I )
DD = T( I, I )
CALL DLANV2( AA, BB, CC, DD, SR( KWTOP+I-2 ),
$ SI( KWTOP+I-2 ), SR( KWTOP+I-1 ),
$ SI( KWTOP+I-1 ), CS, SN )
I = I - 2
END IF
GO TO 60
END IF
*
IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
*
* ==== Reflect spike back into lower triangle ====
*
CALL DCOPY( NS, V, LDV, WORK, 1 )
BETA = WORK( 1 )
CALL DLARFG( NS, BETA, WORK( 2 ), 1, TAU )
WORK( 1 ) = ONE
*
CALL DLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ), LDT )
*
CALL DLARF( 'L', NS, JW, WORK, 1, TAU, T, LDT,
$ WORK( JW+1 ) )
CALL DLARF( 'R', NS, NS, WORK, 1, TAU, T, LDT,
$ WORK( JW+1 ) )
CALL DLARF( 'R', JW, NS, WORK, 1, TAU, V, LDV,
$ WORK( JW+1 ) )
*
CALL DGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
$ LWORK-JW, INFO )
END IF
*
* ==== Copy updated reduced window into place ====
*
IF( KWTOP.GT.1 )
$ H( KWTOP, KWTOP-1 ) = S*V( 1, 1 )
CALL DLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
CALL DCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
$ LDH+1 )
*
* ==== Accumulate orthogonal matrix in order update
* . H and Z, if requested. ====
*
IF( NS.GT.1 .AND. S.NE.ZERO )
$ CALL DORMHR( 'R', 'N', JW, NS, 1, NS, T, LDT, WORK, V, LDV,
$ WORK( JW+1 ), LWORK-JW, INFO )
*
* ==== Update vertical slab in H ====
*
IF( WANTT ) THEN
LTOP = 1
ELSE
LTOP = KTOP
END IF
DO 70 KROW = LTOP, KWTOP - 1, NV
KLN = MIN( NV, KWTOP-KROW )
CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
$ LDH, V, LDV, ZERO, WV, LDWV )
CALL DLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ), LDH )
70 CONTINUE
*
* ==== Update horizontal slab in H ====
*
IF( WANTT ) THEN
DO 80 KCOL = KBOT + 1, N, NH
KLN = MIN( NH, N-KCOL+1 )
CALL DGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
$ H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
CALL DLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
$ LDH )
80 CONTINUE
END IF
*
* ==== Update vertical slab in Z ====
*
IF( WANTZ ) THEN
DO 90 KROW = ILOZ, IHIZ, NV
KLN = MIN( NV, IHIZ-KROW+1 )
CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW, KWTOP ),
$ LDZ, V, LDV, ZERO, WV, LDWV )
CALL DLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
$ LDZ )
90 CONTINUE
END IF
END IF
*
* ==== Return the number of deflations ... ====
*
ND = JW - NS
*
* ==== ... and the number of shifts. (Subtracting
* . INFQR from the spike length takes care
* . of the case of a rare QR failure while
* . calculating eigenvalues of the deflation
* . window.) ====
*
NS = NS - INFQR
*
* ==== Return optimal workspace. ====
*
WORK( 1 ) = DBLE( LWKOPT )
*
* ==== End of DLAQR2 ====
*
END
*> \brief \b DLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR3 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
* IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
* LDT, NV, WV, LDWV, WORK, LWORK )
*
* .. Scalar Arguments ..
* INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
* $ LDZ, LWORK, N, ND, NH, NS, NV, NW
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
* $ V( LDV, * ), WORK( * ), WV( LDWV, * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Aggressive early deflation:
*>
*> DLAQR3 accepts as input an upper Hessenberg matrix
*> H and performs an orthogonal similarity transformation
*> designed to detect and deflate fully converged eigenvalues from
*> a trailing principal submatrix. On output H has been over-
*> written by a new Hessenberg matrix that is a perturbation of
*> an orthogonal similarity transformation of H. It is to be
*> hoped that the final version of H has many zero subdiagonal
*> entries.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> If .TRUE., then the Hessenberg matrix H is fully updated
*> so that the quasi-triangular Schur factor may be
*> computed (in cooperation with the calling subroutine).
*> If .FALSE., then only enough of H is updated to preserve
*> the eigenvalues.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> If .TRUE., then the orthogonal matrix Z is updated so
*> so that the orthogonal Schur factor may be computed
*> (in cooperation with the calling subroutine).
*> If .FALSE., then Z is not referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H and (if WANTZ is .TRUE.) the
*> order of the orthogonal matrix Z.
*> \endverbatim
*>
*> \param[in] KTOP
*> \verbatim
*> KTOP is INTEGER
*> It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
*> KBOT and KTOP together determine an isolated block
*> along the diagonal of the Hessenberg matrix.
*> \endverbatim
*>
*> \param[in] KBOT
*> \verbatim
*> KBOT is INTEGER
*> It is assumed without a check that either
*> KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
*> determine an isolated block along the diagonal of the
*> Hessenberg matrix.
*> \endverbatim
*>
*> \param[in] NW
*> \verbatim
*> NW is INTEGER
*> Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On input the initial N-by-N section of H stores the
*> Hessenberg matrix undergoing aggressive early deflation.
*> On output H has been transformed by an orthogonal
*> similarity transformation, perturbed, and the returned
*> to Hessenberg form that (it is to be hoped) has some
*> zero subdiagonal entries.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is integer
*> Leading dimension of H just as declared in the calling
*> subroutine. N .LE. LDH
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> IF WANTZ is .TRUE., then on output, the orthogonal
*> similarity transformation mentioned above has been
*> accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
*> If WANTZ is .FALSE., then Z is unreferenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is integer
*> The leading dimension of Z just as declared in the
*> calling subroutine. 1 .LE. LDZ.
*> \endverbatim
*>
*> \param[out] NS
*> \verbatim
*> NS is integer
*> The number of unconverged (ie approximate) eigenvalues
*> returned in SR and SI that may be used as shifts by the
*> calling subroutine.
*> \endverbatim
*>
*> \param[out] ND
*> \verbatim
*> ND is integer
*> The number of converged eigenvalues uncovered by this
*> subroutine.
*> \endverbatim
*>
*> \param[out] SR
*> \verbatim
*> SR is DOUBLE PRECISION array, dimension (KBOT)
*> \endverbatim
*>
*> \param[out] SI
*> \verbatim
*> SI is DOUBLE PRECISION array, dimension (KBOT)
*> On output, the real and imaginary parts of approximate
*> eigenvalues that may be used for shifts are stored in
*> SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
*> SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
*> The real and imaginary parts of converged eigenvalues
*> are stored in SR(KBOT-ND+1) through SR(KBOT) and
*> SI(KBOT-ND+1) through SI(KBOT), respectively.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,NW)
*> An NW-by-NW work array.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is integer scalar
*> The leading dimension of V just as declared in the
*> calling subroutine. NW .LE. LDV
*> \endverbatim
*>
*> \param[in] NH
*> \verbatim
*> NH is integer scalar
*> The number of columns of T. NH.GE.NW.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,NW)
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is integer
*> The leading dimension of T just as declared in the
*> calling subroutine. NW .LE. LDT
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*> NV is integer
*> The number of rows of work array WV available for
*> workspace. NV.GE.NW.
*> \endverbatim
*>
*> \param[out] WV
*> \verbatim
*> WV is DOUBLE PRECISION array, dimension (LDWV,NW)
*> \endverbatim
*>
*> \param[in] LDWV
*> \verbatim
*> LDWV is integer
*> The leading dimension of W just as declared in the
*> calling subroutine. NW .LE. LDV
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> On exit, WORK(1) is set to an estimate of the optimal value
*> of LWORK for the given values of N, NW, KTOP and KBOT.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is integer
*> The dimension of the work array WORK. LWORK = 2*NW
*> suffices, but greater efficiency may result from larger
*> values of LWORK.
*>
*> If LWORK = -1, then a workspace query is assumed; DLAQR3
*> only estimates the optimal workspace size for the given
*> values of N, NW, KTOP and KBOT. The estimate is returned
*> in WORK(1). No error message related to LWORK is issued
*> by XERBLA. Neither H nor Z are accessed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*>
* =====================================================================
SUBROUTINE DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
$ IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
$ LDT, NV, WV, LDWV, WORK, LWORK )
*
* -- LAPACK auxiliary routine (version 3.6.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
$ LDZ, LWORK, N, ND, NH, NS, NV, NW
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
$ V( LDV, * ), WORK( * ), WV( LDWV, * ),
$ Z( LDZ, * )
* ..
*
* ================================================================
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, BB, BETA, CC, CS, DD, EVI, EVK, FOO, S,
$ SAFMAX, SAFMIN, SMLNUM, SN, TAU, ULP
INTEGER I, IFST, ILST, INFO, INFQR, J, JW, K, KCOL,
$ KEND, KLN, KROW, KWTOP, LTOP, LWK1, LWK2, LWK3,
$ LWKOPT, NMIN
LOGICAL BULGE, SORTED
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
INTEGER ILAENV
EXTERNAL DLAMCH, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEHRD, DGEMM, DLABAD, DLACPY, DLAHQR,
$ DLANV2, DLAQR4, DLARF, DLARFG, DLASET, DORMHR,
$ DTREXC
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* ==== Estimate optimal workspace. ====
*
JW = MIN( NW, KBOT-KTOP+1 )
IF( JW.LE.2 ) THEN
LWKOPT = 1
ELSE
*
* ==== Workspace query call to DGEHRD ====
*
CALL DGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
LWK1 = INT( WORK( 1 ) )
*
* ==== Workspace query call to DORMHR ====
*
CALL DORMHR( 'R', 'N', JW, JW, 1, JW-1, T, LDT, WORK, V, LDV,
$ WORK, -1, INFO )
LWK2 = INT( WORK( 1 ) )
*
* ==== Workspace query call to DLAQR4 ====
*
CALL DLAQR4( .true., .true., JW, 1, JW, T, LDT, SR, SI, 1, JW,
$ V, LDV, WORK, -1, INFQR )
LWK3 = INT( WORK( 1 ) )
*
* ==== Optimal workspace ====
*
LWKOPT = MAX( JW+MAX( LWK1, LWK2 ), LWK3 )
END IF
*
* ==== Quick return in case of workspace query. ====
*
IF( LWORK.EQ.-1 ) THEN
WORK( 1 ) = DBLE( LWKOPT )
RETURN
END IF
*
* ==== Nothing to do ...
* ... for an empty active block ... ====
NS = 0
ND = 0
WORK( 1 ) = ONE
IF( KTOP.GT.KBOT )
$ RETURN
* ... nor for an empty deflation window. ====
IF( NW.LT.1 )
$ RETURN
*
* ==== Machine constants ====
*
SAFMIN = DLAMCH( 'SAFE MINIMUM' )
SAFMAX = ONE / SAFMIN
CALL DLABAD( SAFMIN, SAFMAX )
ULP = DLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( DBLE( N ) / ULP )
*
* ==== Setup deflation window ====
*
JW = MIN( NW, KBOT-KTOP+1 )
KWTOP = KBOT - JW + 1
IF( KWTOP.EQ.KTOP ) THEN
S = ZERO
ELSE
S = H( KWTOP, KWTOP-1 )
END IF
*
IF( KBOT.EQ.KWTOP ) THEN
*
* ==== 1-by-1 deflation window: not much to do ====
*
SR( KWTOP ) = H( KWTOP, KWTOP )
SI( KWTOP ) = ZERO
NS = 1
ND = 0
IF( ABS( S ).LE.MAX( SMLNUM, ULP*ABS( H( KWTOP, KWTOP ) ) ) )
$ THEN
NS = 0
ND = 1
IF( KWTOP.GT.KTOP )
$ H( KWTOP, KWTOP-1 ) = ZERO
END IF
WORK( 1 ) = ONE
RETURN
END IF
*
* ==== Convert to spike-triangular form. (In case of a
* . rare QR failure, this routine continues to do
* . aggressive early deflation using that part of
* . the deflation window that converged using INFQR
* . here and there to keep track.) ====
*
CALL DLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
CALL DCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ), LDT+1 )
*
CALL DLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
NMIN = ILAENV( 12, 'DLAQR3', 'SV', JW, 1, JW, LWORK )
IF( JW.GT.NMIN ) THEN
CALL DLAQR4( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
$ SI( KWTOP ), 1, JW, V, LDV, WORK, LWORK, INFQR )
ELSE
CALL DLAHQR( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
$ SI( KWTOP ), 1, JW, V, LDV, INFQR )
END IF
*
* ==== DTREXC needs a clean margin near the diagonal ====
*
DO 10 J = 1, JW - 3
T( J+2, J ) = ZERO
T( J+3, J ) = ZERO
10 CONTINUE
IF( JW.GT.2 )
$ T( JW, JW-2 ) = ZERO
*
* ==== Deflation detection loop ====
*
NS = JW
ILST = INFQR + 1
20 CONTINUE
IF( ILST.LE.NS ) THEN
IF( NS.EQ.1 ) THEN
BULGE = .FALSE.
ELSE
BULGE = T( NS, NS-1 ).NE.ZERO
END IF
*
* ==== Small spike tip test for deflation ====
*
IF( .NOT. BULGE ) THEN
*
* ==== Real eigenvalue ====
*
FOO = ABS( T( NS, NS ) )
IF( FOO.EQ.ZERO )
$ FOO = ABS( S )
IF( ABS( S*V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) ) THEN
*
* ==== Deflatable ====
*
NS = NS - 1
ELSE
*
* ==== Undeflatable. Move it up out of the way.
* . (DTREXC can not fail in this case.) ====
*
IFST = NS
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
$ INFO )
ILST = ILST + 1
END IF
ELSE
*
* ==== Complex conjugate pair ====
*
FOO = ABS( T( NS, NS ) ) + SQRT( ABS( T( NS, NS-1 ) ) )*
$ SQRT( ABS( T( NS-1, NS ) ) )
IF( FOO.EQ.ZERO )
$ FOO = ABS( S )
IF( MAX( ABS( S*V( 1, NS ) ), ABS( S*V( 1, NS-1 ) ) ).LE.
$ MAX( SMLNUM, ULP*FOO ) ) THEN
*
* ==== Deflatable ====
*
NS = NS - 2
ELSE
*
* ==== Undeflatable. Move them up out of the way.
* . Fortunately, DTREXC does the right thing with
* . ILST in case of a rare exchange failure. ====
*
IFST = NS
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
$ INFO )
ILST = ILST + 2
END IF
END IF
*
* ==== End deflation detection loop ====
*
GO TO 20
END IF
*
* ==== Return to Hessenberg form ====
*
IF( NS.EQ.0 )
$ S = ZERO
*
IF( NS.LT.JW ) THEN
*
* ==== sorting diagonal blocks of T improves accuracy for
* . graded matrices. Bubble sort deals well with
* . exchange failures. ====
*
SORTED = .false.
I = NS + 1
30 CONTINUE
IF( SORTED )
$ GO TO 50
SORTED = .true.
*
KEND = I - 1
I = INFQR + 1
IF( I.EQ.NS ) THEN
K = I + 1
ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
K = I + 1
ELSE
K = I + 2
END IF
40 CONTINUE
IF( K.LE.KEND ) THEN
IF( K.EQ.I+1 ) THEN
EVI = ABS( T( I, I ) )
ELSE
EVI = ABS( T( I, I ) ) + SQRT( ABS( T( I+1, I ) ) )*
$ SQRT( ABS( T( I, I+1 ) ) )
END IF
*
IF( K.EQ.KEND ) THEN
EVK = ABS( T( K, K ) )
ELSE IF( T( K+1, K ).EQ.ZERO ) THEN
EVK = ABS( T( K, K ) )
ELSE
EVK = ABS( T( K, K ) ) + SQRT( ABS( T( K+1, K ) ) )*
$ SQRT( ABS( T( K, K+1 ) ) )
END IF
*
IF( EVI.GE.EVK ) THEN
I = K
ELSE
SORTED = .false.
IFST = I
ILST = K
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
$ INFO )
IF( INFO.EQ.0 ) THEN
I = ILST
ELSE
I = K
END IF
END IF
IF( I.EQ.KEND ) THEN
K = I + 1
ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
K = I + 1
ELSE
K = I + 2
END IF
GO TO 40
END IF
GO TO 30
50 CONTINUE
END IF
*
* ==== Restore shift/eigenvalue array from T ====
*
I = JW
60 CONTINUE
IF( I.GE.INFQR+1 ) THEN
IF( I.EQ.INFQR+1 ) THEN
SR( KWTOP+I-1 ) = T( I, I )
SI( KWTOP+I-1 ) = ZERO
I = I - 1
ELSE IF( T( I, I-1 ).EQ.ZERO ) THEN
SR( KWTOP+I-1 ) = T( I, I )
SI( KWTOP+I-1 ) = ZERO
I = I - 1
ELSE
AA = T( I-1, I-1 )
CC = T( I, I-1 )
BB = T( I-1, I )
DD = T( I, I )
CALL DLANV2( AA, BB, CC, DD, SR( KWTOP+I-2 ),
$ SI( KWTOP+I-2 ), SR( KWTOP+I-1 ),
$ SI( KWTOP+I-1 ), CS, SN )
I = I - 2
END IF
GO TO 60
END IF
*
IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
*
* ==== Reflect spike back into lower triangle ====
*
CALL DCOPY( NS, V, LDV, WORK, 1 )
BETA = WORK( 1 )
CALL DLARFG( NS, BETA, WORK( 2 ), 1, TAU )
WORK( 1 ) = ONE
*
CALL DLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ), LDT )
*
CALL DLARF( 'L', NS, JW, WORK, 1, TAU, T, LDT,
$ WORK( JW+1 ) )
CALL DLARF( 'R', NS, NS, WORK, 1, TAU, T, LDT,
$ WORK( JW+1 ) )
CALL DLARF( 'R', JW, NS, WORK, 1, TAU, V, LDV,
$ WORK( JW+1 ) )
*
CALL DGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
$ LWORK-JW, INFO )
END IF
*
* ==== Copy updated reduced window into place ====
*
IF( KWTOP.GT.1 )
$ H( KWTOP, KWTOP-1 ) = S*V( 1, 1 )
CALL DLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
CALL DCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
$ LDH+1 )
*
* ==== Accumulate orthogonal matrix in order update
* . H and Z, if requested. ====
*
IF( NS.GT.1 .AND. S.NE.ZERO )
$ CALL DORMHR( 'R', 'N', JW, NS, 1, NS, T, LDT, WORK, V, LDV,
$ WORK( JW+1 ), LWORK-JW, INFO )
*
* ==== Update vertical slab in H ====
*
IF( WANTT ) THEN
LTOP = 1
ELSE
LTOP = KTOP
END IF
DO 70 KROW = LTOP, KWTOP - 1, NV
KLN = MIN( NV, KWTOP-KROW )
CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
$ LDH, V, LDV, ZERO, WV, LDWV )
CALL DLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ), LDH )
70 CONTINUE
*
* ==== Update horizontal slab in H ====
*
IF( WANTT ) THEN
DO 80 KCOL = KBOT + 1, N, NH
KLN = MIN( NH, N-KCOL+1 )
CALL DGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
$ H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
CALL DLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
$ LDH )
80 CONTINUE
END IF
*
* ==== Update vertical slab in Z ====
*
IF( WANTZ ) THEN
DO 90 KROW = ILOZ, IHIZ, NV
KLN = MIN( NV, IHIZ-KROW+1 )
CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW, KWTOP ),
$ LDZ, V, LDV, ZERO, WV, LDWV )
CALL DLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
$ LDZ )
90 CONTINUE
END IF
END IF
*
* ==== Return the number of deflations ... ====
*
ND = JW - NS
*
* ==== ... and the number of shifts. (Subtracting
* . INFQR from the spike length takes care
* . of the case of a rare QR failure while
* . calculating eigenvalues of the deflation
* . window.) ====
*
NS = NS - INFQR
*
* ==== Return optimal workspace. ====
*
WORK( 1 ) = DBLE( LWKOPT )
*
* ==== End of DLAQR3 ====
*
END
*> \brief \b DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR4 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
* ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQR4 implements one level of recursion for DLAQR0.
*> It is a complete implementation of the small bulge multi-shift
*> QR algorithm. It may be called by DLAQR0 and, for large enough
*> deflation window size, it may be called by DLAQR3. This
*> subroutine is identical to DLAQR0 except that it calls DLAQR2
*> instead of DLAQR3.
*>
*> DLAQR4 computes the eigenvalues of a Hessenberg matrix H
*> and, optionally, the matrices T and Z from the Schur decomposition
*> H = Z T Z**T, where T is an upper quasi-triangular matrix (the
*> Schur form), and Z is the orthogonal matrix of Schur vectors.
*>
*> Optionally Z may be postmultiplied into an input orthogonal
*> matrix Q so that this routine can give the Schur factorization
*> of a matrix A which has been reduced to the Hessenberg form H
*> by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> = .TRUE. : the full Schur form T is required;
*> = .FALSE.: only eigenvalues are required.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> = .TRUE. : the matrix of Schur vectors Z is required;
*> = .FALSE.: Schur vectors are not required.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N .GE. 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> It is assumed that H is already upper triangular in rows
*> and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
*> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
*> previous call to DGEBAL, and then passed to DGEHRD when the
*> matrix output by DGEBAL is reduced to Hessenberg form.
*> Otherwise, ILO and IHI should be set to 1 and N,
*> respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
*> If N = 0, then ILO = 1 and IHI = 0.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On entry, the upper Hessenberg matrix H.
*> On exit, if INFO = 0 and WANTT is .TRUE., then H contains
*> the upper quasi-triangular matrix T from the Schur
*> decomposition (the Schur form); 2-by-2 diagonal blocks
*> (corresponding to complex conjugate pairs of eigenvalues)
*> are returned in standard form, with H(i,i) = H(i+1,i+1)
*> and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
*> .FALSE., then the contents of H are unspecified on exit.
*> (The output value of H when INFO.GT.0 is given under the
*> description of INFO below.)
*>
*> This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
*> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH .GE. max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (IHI)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (IHI)
*> The real and imaginary parts, respectively, of the computed
*> eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
*> and WI(ILO:IHI). If two eigenvalues are computed as a
*> complex conjugate pair, they are stored in consecutive
*> elements of WR and WI, say the i-th and (i+1)th, with
*> WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
*> the eigenvalues are stored in the same order as on the
*> diagonal of the Schur form returned in H, with
*> WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
*> block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
*> WI(i+1) = -WI(i).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE..
*> 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,IHI)
*> If WANTZ is .FALSE., then Z is not referenced.
*> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
*> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
*> orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
*> (The output value of Z when INFO.GT.0 is given under
*> the description of INFO below.)
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. if WANTZ is .TRUE.
*> then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension LWORK
*> On exit, if LWORK = -1, WORK(1) returns an estimate of
*> the optimal value for LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK .GE. max(1,N)
*> is sufficient, but LWORK typically as large as 6*N may
*> be required for optimal performance. A workspace query
*> to determine the optimal workspace size is recommended.
*>
*> If LWORK = -1, then DLAQR4 does a workspace query.
*> In this case, DLAQR4 checks the input parameters and
*> estimates the optimal workspace size for the given
*> values of N, ILO and IHI. The estimate is returned
*> in WORK(1). No error message related to LWORK is
*> issued by XERBLA. Neither H nor Z are accessed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> .GT. 0: if INFO = i, DLAQR4 failed to compute all of
*> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
*> and WI contain those eigenvalues which have been
*> successfully computed. (Failures are rare.)
*>
*> If INFO .GT. 0 and WANT is .FALSE., then on exit,
*> the remaining unconverged eigenvalues are the eigen-
*> values of the upper Hessenberg matrix rows and
*> columns ILO through INFO of the final, output
*> value of H.
*>
*> If INFO .GT. 0 and WANTT is .TRUE., then on exit
*>
*> (*) (initial value of H)*U = U*(final value of H)
*>
*> where U is a orthogonal matrix. The final
*> value of H is upper Hessenberg and triangular in
*> rows and columns INFO+1 through IHI.
*>
*> If INFO .GT. 0 and WANTZ is .TRUE., then on exit
*>
*> (final value of Z(ILO:IHI,ILOZ:IHIZ)
*> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
*>
*> where U is the orthogonal matrix in (*) (regard-
*> less of the value of WANTT.)
*>
*> If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
*> accessed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*
*> \par References:
* ================
*>
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
*> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
*> 929--947, 2002.
*> \n
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part II: Aggressive Early Deflation, SIAM Journal
*> of Matrix Analysis, volume 23, pages 948--973, 2002.
*>
* =====================================================================
SUBROUTINE DLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
$ Z( LDZ, * )
* ..
*
* ================================================================
* .. Parameters ..
*
* ==== Matrices of order NTINY or smaller must be processed by
* . DLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
*
* ==== Exceptional deflation windows: try to cure rare
* . slow convergence by varying the size of the
* . deflation window after KEXNW iterations. ====
INTEGER KEXNW
PARAMETER ( KEXNW = 5 )
*
* ==== Exceptional shifts: try to cure rare slow convergence
* . with ad-hoc exceptional shifts every KEXSH iterations.
* . ====
INTEGER KEXSH
PARAMETER ( KEXSH = 6 )
*
* ==== The constants WILK1 and WILK2 are used to form the
* . exceptional shifts. ====
DOUBLE PRECISION WILK1, WILK2
PARAMETER ( WILK1 = 0.75d0, WILK2 = -0.4375d0 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP
INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
$ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
$ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
$ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
LOGICAL SORTED
CHARACTER JBCMPZ*2
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Local Arrays ..
DOUBLE PRECISION ZDUM( 1, 1 )
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DLAHQR, DLANV2, DLAQR2, DLAQR5
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, MAX, MIN, MOD
* ..
* .. Executable Statements ..
INFO = 0
*
* ==== Quick return for N = 0: nothing to do. ====
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = ONE
RETURN
END IF
*
IF( N.LE.NTINY ) THEN
*
* ==== Tiny matrices must use DLAHQR. ====
*
LWKOPT = 1
IF( LWORK.NE.-1 )
$ CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, INFO )
ELSE
*
* ==== Use small bulge multi-shift QR with aggressive early
* . deflation on larger-than-tiny matrices. ====
*
* ==== Hope for the best. ====
*
INFO = 0
*
* ==== Set up job flags for ILAENV. ====
*
IF( WANTT ) THEN
JBCMPZ( 1: 1 ) = 'S'
ELSE
JBCMPZ( 1: 1 ) = 'E'
END IF
IF( WANTZ ) THEN
JBCMPZ( 2: 2 ) = 'V'
ELSE
JBCMPZ( 2: 2 ) = 'N'
END IF
*
* ==== NWR = recommended deflation window size. At this
* . point, N .GT. NTINY = 11, so there is enough
* . subdiagonal workspace for NWR.GE.2 as required.
* . (In fact, there is enough subdiagonal space for
* . NWR.GE.3.) ====
*
NWR = ILAENV( 13, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
* ==== NSR = recommended number of simultaneous shifts.
* . At this point N .GT. NTINY = 11, so there is at
* . enough subdiagonal workspace for NSR to be even
* . and greater than or equal to two as required. ====
*
NSR = ILAENV( 15, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
* ==== Estimate optimal workspace ====
*
* ==== Workspace query call to DLAQR2 ====
*
CALL DLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
$ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
$ N, H, LDH, WORK, -1 )
*
* ==== Optimal workspace = MAX(DLAQR5, DLAQR2) ====
*
LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
*
* ==== Quick return in case of workspace query. ====
*
IF( LWORK.EQ.-1 ) THEN
WORK( 1 ) = DBLE( LWKOPT )
RETURN
END IF
*
* ==== DLAHQR/DLAQR0 crossover point ====
*
NMIN = ILAENV( 12, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NMIN = MAX( NTINY, NMIN )
*
* ==== Nibble crossover point ====
*
NIBBLE = ILAENV( 14, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NIBBLE = MAX( 0, NIBBLE )
*
* ==== Accumulate reflections during ttswp? Use block
* . 2-by-2 structure during matrix-matrix multiply? ====
*
KACC22 = ILAENV( 16, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
KACC22 = MAX( 0, KACC22 )
KACC22 = MIN( 2, KACC22 )
*
* ==== NWMAX = the largest possible deflation window for
* . which there is sufficient workspace. ====
*
NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
NW = NWMAX
*
* ==== NSMAX = the Largest number of simultaneous shifts
* . for which there is sufficient workspace. ====
*
NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
NSMAX = NSMAX - MOD( NSMAX, 2 )
*
* ==== NDFL: an iteration count restarted at deflation. ====
*
NDFL = 1
*
* ==== ITMAX = iteration limit ====
*
ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
*
* ==== Last row and column in the active block ====
*
KBOT = IHI
*
* ==== Main Loop ====
*
DO 80 IT = 1, ITMAX
*
* ==== Done when KBOT falls below ILO ====
*
IF( KBOT.LT.ILO )
$ GO TO 90
*
* ==== Locate active block ====
*
DO 10 K = KBOT, ILO + 1, -1
IF( H( K, K-1 ).EQ.ZERO )
$ GO TO 20
10 CONTINUE
K = ILO
20 CONTINUE
KTOP = K
*
* ==== Select deflation window size:
* . Typical Case:
* . If possible and advisable, nibble the entire
* . active block. If not, use size MIN(NWR,NWMAX)
* . or MIN(NWR+1,NWMAX) depending upon which has
* . the smaller corresponding subdiagonal entry
* . (a heuristic).
* .
* . Exceptional Case:
* . If there have been no deflations in KEXNW or
* . more iterations, then vary the deflation window
* . size. At first, because, larger windows are,
* . in general, more powerful than smaller ones,
* . rapidly increase the window to the maximum possible.
* . Then, gradually reduce the window size. ====
*
NH = KBOT - KTOP + 1
NWUPBD = MIN( NH, NWMAX )
IF( NDFL.LT.KEXNW ) THEN
NW = MIN( NWUPBD, NWR )
ELSE
NW = MIN( NWUPBD, 2*NW )
END IF
IF( NW.LT.NWMAX ) THEN
IF( NW.GE.NH-1 ) THEN
NW = NH
ELSE
KWTOP = KBOT - NW + 1
IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
$ ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
END IF
END IF
IF( NDFL.LT.KEXNW ) THEN
NDEC = -1
ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
NDEC = NDEC + 1
IF( NW-NDEC.LT.2 )
$ NDEC = 0
NW = NW - NDEC
END IF
*
* ==== Aggressive early deflation:
* . split workspace under the subdiagonal into
* . - an nw-by-nw work array V in the lower
* . left-hand-corner,
* . - an NW-by-at-least-NW-but-more-is-better
* . (NW-by-NHO) horizontal work array along
* . the bottom edge,
* . - an at-least-NW-but-more-is-better (NHV-by-NW)
* . vertical work array along the left-hand-edge.
* . ====
*
KV = N - NW + 1
KT = NW + 1
NHO = ( N-NW-1 ) - KT + 1
KWV = NW + 2
NVE = ( N-NW ) - KWV + 1
*
* ==== Aggressive early deflation ====
*
CALL DLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
$ IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
$ NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
$ WORK, LWORK )
*
* ==== Adjust KBOT accounting for new deflations. ====
*
KBOT = KBOT - LD
*
* ==== KS points to the shifts. ====
*
KS = KBOT - LS + 1
*
* ==== Skip an expensive QR sweep if there is a (partly
* . heuristic) reason to expect that many eigenvalues
* . will deflate without it. Here, the QR sweep is
* . skipped if many eigenvalues have just been deflated
* . or if the remaining active block is small.
*
IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
$ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
*
* ==== NS = nominal number of simultaneous shifts.
* . This may be lowered (slightly) if DLAQR2
* . did not provide that many shifts. ====
*
NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
NS = NS - MOD( NS, 2 )
*
* ==== If there have been no deflations
* . in a multiple of KEXSH iterations,
* . then try exceptional shifts.
* . Otherwise use shifts provided by
* . DLAQR2 above or from the eigenvalues
* . of a trailing principal submatrix. ====
*
IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
KS = KBOT - NS + 1
DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
AA = WILK1*SS + H( I, I )
BB = SS
CC = WILK2*SS
DD = AA
CALL DLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ),
$ WR( I ), WI( I ), CS, SN )
30 CONTINUE
IF( KS.EQ.KTOP ) THEN
WR( KS+1 ) = H( KS+1, KS+1 )
WI( KS+1 ) = ZERO
WR( KS ) = WR( KS+1 )
WI( KS ) = WI( KS+1 )
END IF
ELSE
*
* ==== Got NS/2 or fewer shifts? Use DLAHQR
* . on a trailing principal submatrix to
* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
* . there is enough space below the subdiagonal
* . to fit an NS-by-NS scratch array.) ====
*
IF( KBOT-KS+1.LE.NS / 2 ) THEN
KS = KBOT - NS + 1
KT = N - NS + 1
CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH,
$ H( KT, 1 ), LDH )
CALL DLAHQR( .false., .false., NS, 1, NS,
$ H( KT, 1 ), LDH, WR( KS ), WI( KS ),
$ 1, 1, ZDUM, 1, INF )
KS = KS + INF
*
* ==== In case of a rare QR failure use
* . eigenvalues of the trailing 2-by-2
* . principal submatrix. ====
*
IF( KS.GE.KBOT ) THEN
AA = H( KBOT-1, KBOT-1 )
CC = H( KBOT, KBOT-1 )
BB = H( KBOT-1, KBOT )
DD = H( KBOT, KBOT )
CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
$ WI( KBOT-1 ), WR( KBOT ),
$ WI( KBOT ), CS, SN )
KS = KBOT - 1
END IF
END IF
*
IF( KBOT-KS+1.GT.NS ) THEN
*
* ==== Sort the shifts (Helps a little)
* . Bubble sort keeps complex conjugate
* . pairs together. ====
*
SORTED = .false.
DO 50 K = KBOT, KS + 1, -1
IF( SORTED )
$ GO TO 60
SORTED = .true.
DO 40 I = KS, K - 1
IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
$ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
SORTED = .false.
*
SWAP = WR( I )
WR( I ) = WR( I+1 )
WR( I+1 ) = SWAP
*
SWAP = WI( I )
WI( I ) = WI( I+1 )
WI( I+1 ) = SWAP
END IF
40 CONTINUE
50 CONTINUE
60 CONTINUE
END IF
*
* ==== Shuffle shifts into pairs of real shifts
* . and pairs of complex conjugate shifts
* . assuming complex conjugate shifts are
* . already adjacent to one another. (Yes,
* . they are.) ====
*
DO 70 I = KBOT, KS + 2, -2
IF( WI( I ).NE.-WI( I-1 ) ) THEN
*
SWAP = WR( I )
WR( I ) = WR( I-1 )
WR( I-1 ) = WR( I-2 )
WR( I-2 ) = SWAP
*
SWAP = WI( I )
WI( I ) = WI( I-1 )
WI( I-1 ) = WI( I-2 )
WI( I-2 ) = SWAP
END IF
70 CONTINUE
END IF
*
* ==== If there are only two shifts and both are
* . real, then use only one. ====
*
IF( KBOT-KS+1.EQ.2 ) THEN
IF( WI( KBOT ).EQ.ZERO ) THEN
IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
$ ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
WR( KBOT-1 ) = WR( KBOT )
ELSE
WR( KBOT ) = WR( KBOT-1 )
END IF
END IF
END IF
*
* ==== Use up to NS of the the smallest magnatiude
* . shifts. If there aren't NS shifts available,
* . then use them all, possibly dropping one to
* . make the number of shifts even. ====
*
NS = MIN( NS, KBOT-KS+1 )
NS = NS - MOD( NS, 2 )
KS = KBOT - NS + 1
*
* ==== Small-bulge multi-shift QR sweep:
* . split workspace under the subdiagonal into
* . - a KDU-by-KDU work array U in the lower
* . left-hand-corner,
* . - a KDU-by-at-least-KDU-but-more-is-better
* . (KDU-by-NHo) horizontal work array WH along
* . the bottom edge,
* . - and an at-least-KDU-but-more-is-better-by-KDU
* . (NVE-by-KDU) vertical work WV arrow along
* . the left-hand-edge. ====
*
KDU = 3*NS - 3
KU = N - KDU + 1
KWH = KDU + 1
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
KWV = KDU + 4
NVE = N - KDU - KWV + 1
*
* ==== Small-bulge multi-shift QR sweep ====
*
CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
$ WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
$ LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
$ H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
END IF
*
* ==== Note progress (or the lack of it). ====
*
IF( LD.GT.0 ) THEN
NDFL = 1
ELSE
NDFL = NDFL + 1
END IF
*
* ==== End of main loop ====
80 CONTINUE
*
* ==== Iteration limit exceeded. Set INFO to show where
* . the problem occurred and exit. ====
*
INFO = KBOT
90 CONTINUE
END IF
*
* ==== Return the optimal value of LWORK. ====
*
WORK( 1 ) = DBLE( LWKOPT )
*
* ==== End of DLAQR4 ====
*
END
*> \brief \b DLAQR5 performs a single small-bulge multi-shift QR sweep.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR5 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS,
* SR, SI, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U,
* LDU, NV, WV, LDWV, NH, WH, LDWH )
*
* .. Scalar Arguments ..
* INTEGER IHIZ, ILOZ, KACC22, KBOT, KTOP, LDH, LDU, LDV,
* $ LDWH, LDWV, LDZ, N, NH, NSHFTS, NV
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), U( LDU, * ),
* $ V( LDV, * ), WH( LDWH, * ), WV( LDWV, * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQR5, called by DLAQR0, performs a
*> single small-bulge multi-shift QR sweep.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is logical scalar
*> WANTT = .true. if the quasi-triangular Schur factor
*> is being computed. WANTT is set to .false. otherwise.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is logical scalar
*> WANTZ = .true. if the orthogonal Schur factor is being
*> computed. WANTZ is set to .false. otherwise.
*> \endverbatim
*>
*> \param[in] KACC22
*> \verbatim
*> KACC22 is integer with value 0, 1, or 2.
*> Specifies the computation mode of far-from-diagonal
*> orthogonal updates.
*> = 0: DLAQR5 does not accumulate reflections and does not
*> use matrix-matrix multiply to update far-from-diagonal
*> matrix entries.
*> = 1: DLAQR5 accumulates reflections and uses matrix-matrix
*> multiply to update the far-from-diagonal matrix entries.
*> = 2: DLAQR5 accumulates reflections, uses matrix-matrix
*> multiply to update the far-from-diagonal matrix entries,
*> and takes advantage of 2-by-2 block structure during
*> matrix multiplies.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is integer scalar
*> N is the order of the Hessenberg matrix H upon which this
*> subroutine operates.
*> \endverbatim
*>
*> \param[in] KTOP
*> \verbatim
*> KTOP is integer scalar
*> \endverbatim
*>
*> \param[in] KBOT
*> \verbatim
*> KBOT is integer scalar
*> These are the first and last rows and columns of an
*> isolated diagonal block upon which the QR sweep is to be
*> applied. It is assumed without a check that
*> either KTOP = 1 or H(KTOP,KTOP-1) = 0
*> and
*> either KBOT = N or H(KBOT+1,KBOT) = 0.
*> \endverbatim
*>
*> \param[in] NSHFTS
*> \verbatim
*> NSHFTS is integer scalar
*> NSHFTS gives the number of simultaneous shifts. NSHFTS
*> must be positive and even.
*> \endverbatim
*>
*> \param[in,out] SR
*> \verbatim
*> SR is DOUBLE PRECISION array of size (NSHFTS)
*> \endverbatim
*>
*> \param[in,out] SI
*> \verbatim
*> SI is DOUBLE PRECISION array of size (NSHFTS)
*> SR contains the real parts and SI contains the imaginary
*> parts of the NSHFTS shifts of origin that define the
*> multi-shift QR sweep. On output SR and SI may be
*> reordered.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array of size (LDH,N)
*> On input H contains a Hessenberg matrix. On output a
*> multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
*> to the isolated diagonal block in rows and columns KTOP
*> through KBOT.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is integer scalar
*> LDH is the leading dimension of H just as declared in the
*> calling procedure. LDH.GE.MAX(1,N).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array of size (LDZ,IHIZ)
*> If WANTZ = .TRUE., then the QR Sweep orthogonal
*> similarity transformation is accumulated into
*> Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
*> If WANTZ = .FALSE., then Z is unreferenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is integer scalar
*> LDA is the leading dimension of Z just as declared in
*> the calling procedure. LDZ.GE.N.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array of size (LDV,NSHFTS/2)
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is integer scalar
*> LDV is the leading dimension of V as declared in the
*> calling procedure. LDV.GE.3.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array of size
*> (LDU,3*NSHFTS-3)
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is integer scalar
*> LDU is the leading dimension of U just as declared in the
*> in the calling subroutine. LDU.GE.3*NSHFTS-3.
*> \endverbatim
*>
*> \param[in] NH
*> \verbatim
*> NH is integer scalar
*> NH is the number of columns in array WH available for
*> workspace. NH.GE.1.
*> \endverbatim
*>
*> \param[out] WH
*> \verbatim
*> WH is DOUBLE PRECISION array of size (LDWH,NH)
*> \endverbatim
*>
*> \param[in] LDWH
*> \verbatim
*> LDWH is integer scalar
*> Leading dimension of WH just as declared in the
*> calling procedure. LDWH.GE.3*NSHFTS-3.
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*> NV is integer scalar
*> NV is the number of rows in WV agailable for workspace.
*> NV.GE.1.
*> \endverbatim
*>
*> \param[out] WV
*> \verbatim
*> WV is DOUBLE PRECISION array of size
*> (LDWV,3*NSHFTS-3)
*> \endverbatim
*>
*> \param[in] LDWV
*> \verbatim
*> LDWV is integer scalar
*> LDWV is the leading dimension of WV as declared in the
*> in the calling subroutine. LDWV.GE.NV.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*
*> \par References:
* ================
*>
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
*> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
*> 929--947, 2002.
*>
* =====================================================================
SUBROUTINE DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS,
$ SR, SI, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U,
$ LDU, NV, WV, LDWV, NH, WH, LDWH )
*
* -- LAPACK auxiliary routine (version 3.6.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
INTEGER IHIZ, ILOZ, KACC22, KBOT, KTOP, LDH, LDU, LDV,
$ LDWH, LDWV, LDZ, N, NH, NSHFTS, NV
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), U( LDU, * ),
$ V( LDV, * ), WH( LDWH, * ), WV( LDWV, * ),
$ Z( LDZ, * )
* ..
*
* ================================================================
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION ALPHA, BETA, H11, H12, H21, H22, REFSUM,
$ SAFMAX, SAFMIN, SCL, SMLNUM, SWAP, TST1, TST2,
$ ULP
INTEGER I, I2, I4, INCOL, J, J2, J4, JBOT, JCOL, JLEN,
$ JROW, JTOP, K, K1, KDU, KMS, KNZ, KRCOL, KZS,
$ M, M22, MBOT, MEND, MSTART, MTOP, NBMPS, NDCOL,
$ NS, NU
LOGICAL ACCUM, BLK22, BMP22
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
*
INTRINSIC ABS, DBLE, MAX, MIN, MOD
* ..
* .. Local Arrays ..
DOUBLE PRECISION VT( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLABAD, DLACPY, DLAQR1, DLARFG, DLASET,
$ DTRMM
* ..
* .. Executable Statements ..
*
* ==== If there are no shifts, then there is nothing to do. ====
*
IF( NSHFTS.LT.2 )
$ RETURN
*
* ==== If the active block is empty or 1-by-1, then there
* . is nothing to do. ====
*
IF( KTOP.GE.KBOT )
$ RETURN
*
* ==== Shuffle shifts into pairs of real shifts and pairs
* . of complex conjugate shifts assuming complex
* . conjugate shifts are already adjacent to one
* . another. ====
*
DO 10 I = 1, NSHFTS - 2, 2
IF( SI( I ).NE.-SI( I+1 ) ) THEN
*
SWAP = SR( I )
SR( I ) = SR( I+1 )
SR( I+1 ) = SR( I+2 )
SR( I+2 ) = SWAP
*
SWAP = SI( I )
SI( I ) = SI( I+1 )
SI( I+1 ) = SI( I+2 )
SI( I+2 ) = SWAP
END IF
10 CONTINUE
*
* ==== NSHFTS is supposed to be even, but if it is odd,
* . then simply reduce it by one. The shuffle above
* . ensures that the dropped shift is real and that
* . the remaining shifts are paired. ====
*
NS = NSHFTS - MOD( NSHFTS, 2 )
*
* ==== Machine constants for deflation ====
*
SAFMIN = DLAMCH( 'SAFE MINIMUM' )
SAFMAX = ONE / SAFMIN
CALL DLABAD( SAFMIN, SAFMAX )
ULP = DLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( DBLE( N ) / ULP )
*
* ==== Use accumulated reflections to update far-from-diagonal
* . entries ? ====
*
ACCUM = ( KACC22.EQ.1 ) .OR. ( KACC22.EQ.2 )
*
* ==== If so, exploit the 2-by-2 block structure? ====
*
BLK22 = ( NS.GT.2 ) .AND. ( KACC22.EQ.2 )
*
* ==== clear trash ====
*
IF( KTOP+2.LE.KBOT )
$ H( KTOP+2, KTOP ) = ZERO
*
* ==== NBMPS = number of 2-shift bulges in the chain ====
*
NBMPS = NS / 2
*
* ==== KDU = width of slab ====
*
KDU = 6*NBMPS - 3
*
* ==== Create and chase chains of NBMPS bulges ====
*
DO 220 INCOL = 3*( 1-NBMPS ) + KTOP - 1, KBOT - 2, 3*NBMPS - 2
NDCOL = INCOL + KDU
IF( ACCUM )
$ CALL DLASET( 'ALL', KDU, KDU, ZERO, ONE, U, LDU )
*
* ==== Near-the-diagonal bulge chase. The following loop
* . performs the near-the-diagonal part of a small bulge
* . multi-shift QR sweep. Each 6*NBMPS-2 column diagonal
* . chunk extends from column INCOL to column NDCOL
* . (including both column INCOL and column NDCOL). The
* . following loop chases a 3*NBMPS column long chain of
* . NBMPS bulges 3*NBMPS-2 columns to the right. (INCOL
* . may be less than KTOP and and NDCOL may be greater than
* . KBOT indicating phantom columns from which to chase
* . bulges before they are actually introduced or to which
* . to chase bulges beyond column KBOT.) ====
*
DO 150 KRCOL = INCOL, MIN( INCOL+3*NBMPS-3, KBOT-2 )
*
* ==== Bulges number MTOP to MBOT are active double implicit
* . shift bulges. There may or may not also be small
* . 2-by-2 bulge, if there is room. The inactive bulges
* . (if any) must wait until the active bulges have moved
* . down the diagonal to make room. The phantom matrix
* . paradigm described above helps keep track. ====
*
MTOP = MAX( 1, ( ( KTOP-1 )-KRCOL+2 ) / 3+1 )
MBOT = MIN( NBMPS, ( KBOT-KRCOL ) / 3 )
M22 = MBOT + 1
BMP22 = ( MBOT.LT.NBMPS ) .AND. ( KRCOL+3*( M22-1 ) ).EQ.
$ ( KBOT-2 )
*
* ==== Generate reflections to chase the chain right
* . one column. (The minimum value of K is KTOP-1.) ====
*
DO 20 M = MTOP, MBOT
K = KRCOL + 3*( M-1 )
IF( K.EQ.KTOP-1 ) THEN
CALL DLAQR1( 3, H( KTOP, KTOP ), LDH, SR( 2*M-1 ),
$ SI( 2*M-1 ), SR( 2*M ), SI( 2*M ),
$ V( 1, M ) )
ALPHA = V( 1, M )
CALL DLARFG( 3, ALPHA, V( 2, M ), 1, V( 1, M ) )
ELSE
BETA = H( K+1, K )
V( 2, M ) = H( K+2, K )
V( 3, M ) = H( K+3, K )
CALL DLARFG( 3, BETA, V( 2, M ), 1, V( 1, M ) )
*
* ==== A Bulge may collapse because of vigilant
* . deflation or destructive underflow. In the
* . underflow case, try the two-small-subdiagonals
* . trick to try to reinflate the bulge. ====
*
IF( H( K+3, K ).NE.ZERO .OR. H( K+3, K+1 ).NE.
$ ZERO .OR. H( K+3, K+2 ).EQ.ZERO ) THEN
*
* ==== Typical case: not collapsed (yet). ====
*
H( K+1, K ) = BETA
H( K+2, K ) = ZERO
H( K+3, K ) = ZERO
ELSE
*
* ==== Atypical case: collapsed. Attempt to
* . reintroduce ignoring H(K+1,K) and H(K+2,K).
* . If the fill resulting from the new
* . reflector is too large, then abandon it.
* . Otherwise, use the new one. ====
*
CALL DLAQR1( 3, H( K+1, K+1 ), LDH, SR( 2*M-1 ),
$ SI( 2*M-1 ), SR( 2*M ), SI( 2*M ),
$ VT )
ALPHA = VT( 1 )
CALL DLARFG( 3, ALPHA, VT( 2 ), 1, VT( 1 ) )
REFSUM = VT( 1 )*( H( K+1, K )+VT( 2 )*
$ H( K+2, K ) )
*
IF( ABS( H( K+2, K )-REFSUM*VT( 2 ) )+
$ ABS( REFSUM*VT( 3 ) ).GT.ULP*
$ ( ABS( H( K, K ) )+ABS( H( K+1,
$ K+1 ) )+ABS( H( K+2, K+2 ) ) ) ) THEN
*
* ==== Starting a new bulge here would
* . create non-negligible fill. Use
* . the old one with trepidation. ====
*
H( K+1, K ) = BETA
H( K+2, K ) = ZERO
H( K+3, K ) = ZERO
ELSE
*
* ==== Stating a new bulge here would
* . create only negligible fill.
* . Replace the old reflector with
* . the new one. ====
*
H( K+1, K ) = H( K+1, K ) - REFSUM
H( K+2, K ) = ZERO
H( K+3, K ) = ZERO
V( 1, M ) = VT( 1 )
V( 2, M ) = VT( 2 )
V( 3, M ) = VT( 3 )
END IF
END IF
END IF
20 CONTINUE
*
* ==== Generate a 2-by-2 reflection, if needed. ====
*
K = KRCOL + 3*( M22-1 )
IF( BMP22 ) THEN
IF( K.EQ.KTOP-1 ) THEN
CALL DLAQR1( 2, H( K+1, K+1 ), LDH, SR( 2*M22-1 ),
$ SI( 2*M22-1 ), SR( 2*M22 ), SI( 2*M22 ),
$ V( 1, M22 ) )
BETA = V( 1, M22 )
CALL DLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
ELSE
BETA = H( K+1, K )
V( 2, M22 ) = H( K+2, K )
CALL DLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
H( K+1, K ) = BETA
H( K+2, K ) = ZERO
END IF
END IF
*
* ==== Multiply H by reflections from the left ====
*
IF( ACCUM ) THEN
JBOT = MIN( NDCOL, KBOT )
ELSE IF( WANTT ) THEN
JBOT = N
ELSE
JBOT = KBOT
END IF
DO 40 J = MAX( KTOP, KRCOL ), JBOT
MEND = MIN( MBOT, ( J-KRCOL+2 ) / 3 )
DO 30 M = MTOP, MEND
K = KRCOL + 3*( M-1 )
REFSUM = V( 1, M )*( H( K+1, J )+V( 2, M )*
$ H( K+2, J )+V( 3, M )*H( K+3, J ) )
H( K+1, J ) = H( K+1, J ) - REFSUM
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M )
H( K+3, J ) = H( K+3, J ) - REFSUM*V( 3, M )
30 CONTINUE
40 CONTINUE
IF( BMP22 ) THEN
K = KRCOL + 3*( M22-1 )
DO 50 J = MAX( K+1, KTOP ), JBOT
REFSUM = V( 1, M22 )*( H( K+1, J )+V( 2, M22 )*
$ H( K+2, J ) )
H( K+1, J ) = H( K+1, J ) - REFSUM
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M22 )
50 CONTINUE
END IF
*
* ==== Multiply H by reflections from the right.
* . Delay filling in the last row until the
* . vigilant deflation check is complete. ====
*
IF( ACCUM ) THEN
JTOP = MAX( KTOP, INCOL )
ELSE IF( WANTT ) THEN
JTOP = 1
ELSE
JTOP = KTOP
END IF
DO 90 M = MTOP, MBOT
IF( V( 1, M ).NE.ZERO ) THEN
K = KRCOL + 3*( M-1 )
DO 60 J = JTOP, MIN( KBOT, K+3 )
REFSUM = V( 1, M )*( H( J, K+1 )+V( 2, M )*
$ H( J, K+2 )+V( 3, M )*H( J, K+3 ) )
H( J, K+1 ) = H( J, K+1 ) - REFSUM
H( J, K+2 ) = H( J, K+2 ) - REFSUM*V( 2, M )
H( J, K+3 ) = H( J, K+3 ) - REFSUM*V( 3, M )
60 CONTINUE
*
IF( ACCUM ) THEN
*
* ==== Accumulate U. (If necessary, update Z later
* . with with an efficient matrix-matrix
* . multiply.) ====
*
KMS = K - INCOL
DO 70 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = V( 1, M )*( U( J, KMS+1 )+V( 2, M )*
$ U( J, KMS+2 )+V( 3, M )*U( J, KMS+3 ) )
U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM
U( J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*V( 2, M )
U( J, KMS+3 ) = U( J, KMS+3 ) - REFSUM*V( 3, M )
70 CONTINUE
ELSE IF( WANTZ ) THEN
*
* ==== U is not accumulated, so update Z
* . now by multiplying by reflections
* . from the right. ====
*
DO 80 J = ILOZ, IHIZ
REFSUM = V( 1, M )*( Z( J, K+1 )+V( 2, M )*
$ Z( J, K+2 )+V( 3, M )*Z( J, K+3 ) )
Z( J, K+1 ) = Z( J, K+1 ) - REFSUM
Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*V( 2, M )
Z( J, K+3 ) = Z( J, K+3 ) - REFSUM*V( 3, M )
80 CONTINUE
END IF
END IF
90 CONTINUE
*
* ==== Special case: 2-by-2 reflection (if needed) ====
*
K = KRCOL + 3*( M22-1 )
IF( BMP22 ) THEN
IF ( V( 1, M22 ).NE.ZERO ) THEN
DO 100 J = JTOP, MIN( KBOT, K+3 )
REFSUM = V( 1, M22 )*( H( J, K+1 )+V( 2, M22 )*
$ H( J, K+2 ) )
H( J, K+1 ) = H( J, K+1 ) - REFSUM
H( J, K+2 ) = H( J, K+2 ) - REFSUM*V( 2, M22 )
100 CONTINUE
*
IF( ACCUM ) THEN
KMS = K - INCOL
DO 110 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = V( 1, M22 )*( U( J, KMS+1 )+
$ V( 2, M22 )*U( J, KMS+2 ) )
U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM
U( J, KMS+2 ) = U( J, KMS+2 ) -
$ REFSUM*V( 2, M22 )
110 CONTINUE
ELSE IF( WANTZ ) THEN
DO 120 J = ILOZ, IHIZ
REFSUM = V( 1, M22 )*( Z( J, K+1 )+V( 2, M22 )*
$ Z( J, K+2 ) )
Z( J, K+1 ) = Z( J, K+1 ) - REFSUM
Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*V( 2, M22 )
120 CONTINUE
END IF
END IF
END IF
*
* ==== Vigilant deflation check ====
*
MSTART = MTOP
IF( KRCOL+3*( MSTART-1 ).LT.KTOP )
$ MSTART = MSTART + 1
MEND = MBOT
IF( BMP22 )
$ MEND = MEND + 1
IF( KRCOL.EQ.KBOT-2 )
$ MEND = MEND + 1
DO 130 M = MSTART, MEND
K = MIN( KBOT-1, KRCOL+3*( M-1 ) )
*
* ==== The following convergence test requires that
* . the tradition small-compared-to-nearby-diagonals
* . criterion and the Ahues & Tisseur (LAWN 122, 1997)
* . criteria both be satisfied. The latter improves
* . accuracy in some examples. Falling back on an
* . alternate convergence criterion when TST1 or TST2
* . is zero (as done here) is traditional but probably
* . unnecessary. ====
*
IF( H( K+1, K ).NE.ZERO ) THEN
TST1 = ABS( H( K, K ) ) + ABS( H( K+1, K+1 ) )
IF( TST1.EQ.ZERO ) THEN
IF( K.GE.KTOP+1 )
$ TST1 = TST1 + ABS( H( K, K-1 ) )
IF( K.GE.KTOP+2 )
$ TST1 = TST1 + ABS( H( K, K-2 ) )
IF( K.GE.KTOP+3 )
$ TST1 = TST1 + ABS( H( K, K-3 ) )
IF( K.LE.KBOT-2 )
$ TST1 = TST1 + ABS( H( K+2, K+1 ) )
IF( K.LE.KBOT-3 )
$ TST1 = TST1 + ABS( H( K+3, K+1 ) )
IF( K.LE.KBOT-4 )
$ TST1 = TST1 + ABS( H( K+4, K+1 ) )
END IF
IF( ABS( H( K+1, K ) ).LE.MAX( SMLNUM, ULP*TST1 ) )
$ THEN
H12 = MAX( ABS( H( K+1, K ) ), ABS( H( K, K+1 ) ) )
H21 = MIN( ABS( H( K+1, K ) ), ABS( H( K, K+1 ) ) )
H11 = MAX( ABS( H( K+1, K+1 ) ),
$ ABS( H( K, K )-H( K+1, K+1 ) ) )
H22 = MIN( ABS( H( K+1, K+1 ) ),
$ ABS( H( K, K )-H( K+1, K+1 ) ) )
SCL = H11 + H12
TST2 = H22*( H11 / SCL )
*
IF( TST2.EQ.ZERO .OR. H21*( H12 / SCL ).LE.
$ MAX( SMLNUM, ULP*TST2 ) )H( K+1, K ) = ZERO
END IF
END IF
130 CONTINUE
*
* ==== Fill in the last row of each bulge. ====
*
MEND = MIN( NBMPS, ( KBOT-KRCOL-1 ) / 3 )
DO 140 M = MTOP, MEND
K = KRCOL + 3*( M-1 )
REFSUM = V( 1, M )*V( 3, M )*H( K+4, K+3 )
H( K+4, K+1 ) = -REFSUM
H( K+4, K+2 ) = -REFSUM*V( 2, M )
H( K+4, K+3 ) = H( K+4, K+3 ) - REFSUM*V( 3, M )
140 CONTINUE
*
* ==== End of near-the-diagonal bulge chase. ====
*
150 CONTINUE
*
* ==== Use U (if accumulated) to update far-from-diagonal
* . entries in H. If required, use U to update Z as
* . well. ====
*
IF( ACCUM ) THEN
IF( WANTT ) THEN
JTOP = 1
JBOT = N
ELSE
JTOP = KTOP
JBOT = KBOT
END IF
IF( ( .NOT.BLK22 ) .OR. ( INCOL.LT.KTOP ) .OR.
$ ( NDCOL.GT.KBOT ) .OR. ( NS.LE.2 ) ) THEN
*
* ==== Updates not exploiting the 2-by-2 block
* . structure of U. K1 and NU keep track of
* . the location and size of U in the special
* . cases of introducing bulges and chasing
* . bulges off the bottom. In these special
* . cases and in case the number of shifts
* . is NS = 2, there is no 2-by-2 block
* . structure to exploit. ====
*
K1 = MAX( 1, KTOP-INCOL )
NU = ( KDU-MAX( 0, NDCOL-KBOT ) ) - K1 + 1
*
* ==== Horizontal Multiply ====
*
DO 160 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH
JLEN = MIN( NH, JBOT-JCOL+1 )
CALL DGEMM( 'C', 'N', NU, JLEN, NU, ONE, U( K1, K1 ),
$ LDU, H( INCOL+K1, JCOL ), LDH, ZERO, WH,
$ LDWH )
CALL DLACPY( 'ALL', NU, JLEN, WH, LDWH,
$ H( INCOL+K1, JCOL ), LDH )
160 CONTINUE
*
* ==== Vertical multiply ====
*
DO 170 JROW = JTOP, MAX( KTOP, INCOL ) - 1, NV
JLEN = MIN( NV, MAX( KTOP, INCOL )-JROW )
CALL DGEMM( 'N', 'N', JLEN, NU, NU, ONE,
$ H( JROW, INCOL+K1 ), LDH, U( K1, K1 ),
$ LDU, ZERO, WV, LDWV )
CALL DLACPY( 'ALL', JLEN, NU, WV, LDWV,
$ H( JROW, INCOL+K1 ), LDH )
170 CONTINUE
*
* ==== Z multiply (also vertical) ====
*
IF( WANTZ ) THEN
DO 180 JROW = ILOZ, IHIZ, NV
JLEN = MIN( NV, IHIZ-JROW+1 )
CALL DGEMM( 'N', 'N', JLEN, NU, NU, ONE,
$ Z( JROW, INCOL+K1 ), LDZ, U( K1, K1 ),
$ LDU, ZERO, WV, LDWV )
CALL DLACPY( 'ALL', JLEN, NU, WV, LDWV,
$ Z( JROW, INCOL+K1 ), LDZ )
180 CONTINUE
END IF
ELSE
*
* ==== Updates exploiting U's 2-by-2 block structure.
* . (I2, I4, J2, J4 are the last rows and columns
* . of the blocks.) ====
*
I2 = ( KDU+1 ) / 2
I4 = KDU
J2 = I4 - I2
J4 = KDU
*
* ==== KZS and KNZ deal with the band of zeros
* . along the diagonal of one of the triangular
* . blocks. ====
*
KZS = ( J4-J2 ) - ( NS+1 )
KNZ = NS + 1
*
* ==== Horizontal multiply ====
*
DO 190 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH
JLEN = MIN( NH, JBOT-JCOL+1 )
*
* ==== Copy bottom of H to top+KZS of scratch ====
* (The first KZS rows get multiplied by zero.) ====
*
CALL DLACPY( 'ALL', KNZ, JLEN, H( INCOL+1+J2, JCOL ),
$ LDH, WH( KZS+1, 1 ), LDWH )
*
* ==== Multiply by U21**T ====
*
CALL DLASET( 'ALL', KZS, JLEN, ZERO, ZERO, WH, LDWH )
CALL DTRMM( 'L', 'U', 'C', 'N', KNZ, JLEN, ONE,
$ U( J2+1, 1+KZS ), LDU, WH( KZS+1, 1 ),
$ LDWH )
*
* ==== Multiply top of H by U11**T ====
*
CALL DGEMM( 'C', 'N', I2, JLEN, J2, ONE, U, LDU,
$ H( INCOL+1, JCOL ), LDH, ONE, WH, LDWH )
*
* ==== Copy top of H to bottom of WH ====
*
CALL DLACPY( 'ALL', J2, JLEN, H( INCOL+1, JCOL ), LDH,
$ WH( I2+1, 1 ), LDWH )
*
* ==== Multiply by U21**T ====
*
CALL DTRMM( 'L', 'L', 'C', 'N', J2, JLEN, ONE,
$ U( 1, I2+1 ), LDU, WH( I2+1, 1 ), LDWH )
*
* ==== Multiply by U22 ====
*
CALL DGEMM( 'C', 'N', I4-I2, JLEN, J4-J2, ONE,
$ U( J2+1, I2+1 ), LDU,
$ H( INCOL+1+J2, JCOL ), LDH, ONE,
$ WH( I2+1, 1 ), LDWH )
*
* ==== Copy it back ====
*
CALL DLACPY( 'ALL', KDU, JLEN, WH, LDWH,
$ H( INCOL+1, JCOL ), LDH )
190 CONTINUE
*
* ==== Vertical multiply ====
*
DO 200 JROW = JTOP, MAX( INCOL, KTOP ) - 1, NV
JLEN = MIN( NV, MAX( INCOL, KTOP )-JROW )
*
* ==== Copy right of H to scratch (the first KZS
* . columns get multiplied by zero) ====
*
CALL DLACPY( 'ALL', JLEN, KNZ, H( JROW, INCOL+1+J2 ),
$ LDH, WV( 1, 1+KZS ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL DLASET( 'ALL', JLEN, KZS, ZERO, ZERO, WV, LDWV )
CALL DTRMM( 'R', 'U', 'N', 'N', JLEN, KNZ, ONE,
$ U( J2+1, 1+KZS ), LDU, WV( 1, 1+KZS ),
$ LDWV )
*
* ==== Multiply by U11 ====
*
CALL DGEMM( 'N', 'N', JLEN, I2, J2, ONE,
$ H( JROW, INCOL+1 ), LDH, U, LDU, ONE, WV,
$ LDWV )
*
* ==== Copy left of H to right of scratch ====
*
CALL DLACPY( 'ALL', JLEN, J2, H( JROW, INCOL+1 ), LDH,
$ WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL DTRMM( 'R', 'L', 'N', 'N', JLEN, I4-I2, ONE,
$ U( 1, I2+1 ), LDU, WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U22 ====
*
CALL DGEMM( 'N', 'N', JLEN, I4-I2, J4-J2, ONE,
$ H( JROW, INCOL+1+J2 ), LDH,
$ U( J2+1, I2+1 ), LDU, ONE, WV( 1, 1+I2 ),
$ LDWV )
*
* ==== Copy it back ====
*
CALL DLACPY( 'ALL', JLEN, KDU, WV, LDWV,
$ H( JROW, INCOL+1 ), LDH )
200 CONTINUE
*
* ==== Multiply Z (also vertical) ====
*
IF( WANTZ ) THEN
DO 210 JROW = ILOZ, IHIZ, NV
JLEN = MIN( NV, IHIZ-JROW+1 )
*
* ==== Copy right of Z to left of scratch (first
* . KZS columns get multiplied by zero) ====
*
CALL DLACPY( 'ALL', JLEN, KNZ,
$ Z( JROW, INCOL+1+J2 ), LDZ,
$ WV( 1, 1+KZS ), LDWV )
*
* ==== Multiply by U12 ====
*
CALL DLASET( 'ALL', JLEN, KZS, ZERO, ZERO, WV,
$ LDWV )
CALL DTRMM( 'R', 'U', 'N', 'N', JLEN, KNZ, ONE,
$ U( J2+1, 1+KZS ), LDU, WV( 1, 1+KZS ),
$ LDWV )
*
* ==== Multiply by U11 ====
*
CALL DGEMM( 'N', 'N', JLEN, I2, J2, ONE,
$ Z( JROW, INCOL+1 ), LDZ, U, LDU, ONE,
$ WV, LDWV )
*
* ==== Copy left of Z to right of scratch ====
*
CALL DLACPY( 'ALL', JLEN, J2, Z( JROW, INCOL+1 ),
$ LDZ, WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL DTRMM( 'R', 'L', 'N', 'N', JLEN, I4-I2, ONE,
$ U( 1, I2+1 ), LDU, WV( 1, 1+I2 ),
$ LDWV )
*
* ==== Multiply by U22 ====
*
CALL DGEMM( 'N', 'N', JLEN, I4-I2, J4-J2, ONE,
$ Z( JROW, INCOL+1+J2 ), LDZ,
$ U( J2+1, I2+1 ), LDU, ONE,
$ WV( 1, 1+I2 ), LDWV )
*
* ==== Copy the result back to Z ====
*
CALL DLACPY( 'ALL', JLEN, KDU, WV, LDWV,
$ Z( JROW, INCOL+1 ), LDZ )
210 CONTINUE
END IF
END IF
END IF
220 CONTINUE
*
* ==== End of DLAQR5 ====
*
END
*> \brief \b DLARF applies an elementary reflector to a general rectangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARF + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARF( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
* .. Scalar Arguments ..
* CHARACTER SIDE
* INTEGER INCV, LDC, M, N
* DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARF applies a real elementary reflector H to a real m by n matrix
*> C, from either the left or the right. H is represented in the form
*>
*> H = I - tau * v * v**T
*>
*> where tau is a real scalar and v is a real vector.
*>
*> If tau = 0, then H is taken to be the unit matrix.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': form H * C
*> = 'R': form C * H
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (1 + (M-1)*abs(INCV)) if SIDE = 'L'
*> or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
*> The vector v in the representation of H. V is not used if
*> TAU = 0.
*> \endverbatim
*>
*> \param[in] INCV
*> \verbatim
*> INCV is INTEGER
*> The increment between elements of v. INCV <> 0.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The value tau in the representation of H.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by the matrix H * C if SIDE = 'L',
*> or C * H if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L'
*> or (M) if SIDE = 'R'
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLARF( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER SIDE
INTEGER INCV, LDC, M, N
DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL APPLYLEFT
INTEGER I, LASTV, LASTC
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DGER
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILADLR, ILADLC
EXTERNAL LSAME, ILADLR, ILADLC
* ..
* .. Executable Statements ..
*
APPLYLEFT = LSAME( SIDE, 'L' )
LASTV = 0
LASTC = 0
IF( TAU.NE.ZERO ) THEN
! Set up variables for scanning V. LASTV begins pointing to the end
! of V.
IF( APPLYLEFT ) THEN
LASTV = M
ELSE
LASTV = N
END IF
IF( INCV.GT.0 ) THEN
I = 1 + (LASTV-1) * INCV
ELSE
I = 1
END IF
! Look for the last non-zero row in V.
DO WHILE( LASTV.GT.0 .AND. V( I ).EQ.ZERO )
LASTV = LASTV - 1
I = I - INCV
END DO
IF( APPLYLEFT ) THEN
! Scan for the last non-zero column in C(1:lastv,:).
LASTC = ILADLC(LASTV, N, C, LDC)
ELSE
! Scan for the last non-zero row in C(:,1:lastv).
LASTC = ILADLR(M, LASTV, C, LDC)
END IF
END IF
! Note that lastc.eq.0 renders the BLAS operations null; no special
! case is needed at this level.
IF( APPLYLEFT ) THEN
*
* Form H * C
*
IF( LASTV.GT.0 ) THEN
*
* w(1:lastc,1) := C(1:lastv,1:lastc)**T * v(1:lastv,1)
*
CALL DGEMV( 'Transpose', LASTV, LASTC, ONE, C, LDC, V, INCV,
$ ZERO, WORK, 1 )
*
* C(1:lastv,1:lastc) := C(...) - v(1:lastv,1) * w(1:lastc,1)**T
*
CALL DGER( LASTV, LASTC, -TAU, V, INCV, WORK, 1, C, LDC )
END IF
ELSE
*
* Form C * H
*
IF( LASTV.GT.0 ) THEN
*
* w(1:lastc,1) := C(1:lastc,1:lastv) * v(1:lastv,1)
*
CALL DGEMV( 'No transpose', LASTC, LASTV, ONE, C, LDC,
$ V, INCV, ZERO, WORK, 1 )
*
* C(1:lastc,1:lastv) := C(...) - w(1:lastc,1) * v(1:lastv,1)**T
*
CALL DGER( LASTC, LASTV, -TAU, WORK, 1, V, INCV, C, LDC )
END IF
END IF
RETURN
*
* End of DLARF
*
END
*> \brief \b DLARFB applies a block reflector or its transpose to a general rectangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFB + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV,
* T, LDT, C, LDC, WORK, LDWORK )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, SIDE, STOREV, TRANS
* INTEGER K, LDC, LDT, LDV, LDWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), T( LDT, * ), V( LDV, * ),
* $ WORK( LDWORK, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARFB applies a real block reflector H or its transpose H**T to a
*> real m by n matrix C, from either the left or the right.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply H or H**T from the Left
*> = 'R': apply H or H**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': apply H (No transpose)
*> = 'T': apply H**T (Transpose)
*> \endverbatim
*>
*> \param[in] DIRECT
*> \verbatim
*> DIRECT is CHARACTER*1
*> Indicates how H is formed from a product of elementary
*> reflectors
*> = 'F': H = H(1) H(2) . . . H(k) (Forward)
*> = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*> STOREV is CHARACTER*1
*> Indicates how the vectors which define the elementary
*> reflectors are stored:
*> = 'C': Columnwise
*> = 'R': Rowwise
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The order of the matrix T (= the number of elementary
*> reflectors whose product defines the block reflector).
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (LDV,K) if STOREV = 'C'
*> (LDV,M) if STOREV = 'R' and SIDE = 'L'
*> (LDV,N) if STOREV = 'R' and SIDE = 'R'
*> The matrix V. See Further Details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V.
*> If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
*> if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
*> if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,K)
*> The triangular k by k matrix T in the representation of the
*> block reflector.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= K.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LDWORK,K)
*> \endverbatim
*>
*> \param[in] LDWORK
*> \verbatim
*> LDWORK is INTEGER
*> The leading dimension of the array WORK.
*> If SIDE = 'L', LDWORK >= max(1,N);
*> if SIDE = 'R', LDWORK >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2013
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The shape of the matrix V and the storage of the vectors which define
*> the H(i) is best illustrated by the following example with n = 5 and
*> k = 3. The elements equal to 1 are not stored; the corresponding
*> array elements are modified but restored on exit. The rest of the
*> array is not used.
*>
*> DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
*>
*> V = ( 1 ) V = ( 1 v1 v1 v1 v1 )
*> ( v1 1 ) ( 1 v2 v2 v2 )
*> ( v1 v2 1 ) ( 1 v3 v3 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*>
*> DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
*>
*> V = ( v1 v2 v3 ) V = ( v1 v1 1 )
*> ( v1 v2 v3 ) ( v2 v2 v2 1 )
*> ( 1 v2 v3 ) ( v3 v3 v3 v3 1 )
*> ( 1 v3 )
*> ( 1 )
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV,
$ T, LDT, C, LDC, WORK, LDWORK )
*
* -- LAPACK auxiliary routine (version 3.5.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2013
*
* .. Scalar Arguments ..
CHARACTER DIRECT, SIDE, STOREV, TRANS
INTEGER K, LDC, LDT, LDV, LDWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), T( LDT, * ), V( LDV, * ),
$ WORK( LDWORK, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
CHARACTER TRANST
INTEGER I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DTRMM
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.LE.0 .OR. N.LE.0 )
$ RETURN
*
IF( LSAME( TRANS, 'N' ) ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
IF( LSAME( STOREV, 'C' ) ) THEN
*
IF( LSAME( DIRECT, 'F' ) ) THEN
*
* Let V = ( V1 ) (first K rows)
* ( V2 )
* where V1 is unit lower triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H**T * C where C = ( C1 )
* ( C2 )
*
* W := C**T * V = (C1**T * V1 + C2**T * V2) (stored in WORK)
*
* W := C1**T
*
DO 10 J = 1, K
CALL DCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 )
10 CONTINUE
*
* W := W * V1
*
CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit', N,
$ K, ONE, V, LDV, WORK, LDWORK )
IF( M.GT.K ) THEN
*
* W := W + C2**T * V2
*
CALL DGEMM( 'Transpose', 'No transpose', N, K, M-K,
$ ONE, C( K+1, 1 ), LDC, V( K+1, 1 ), LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T**T or W * T
*
CALL DTRMM( 'Right', 'Upper', TRANST, 'Non-unit', N, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - V * W**T
*
IF( M.GT.K ) THEN
*
* C2 := C2 - V2 * W**T
*
CALL DGEMM( 'No transpose', 'Transpose', M-K, N, K,
$ -ONE, V( K+1, 1 ), LDV, WORK, LDWORK, ONE,
$ C( K+1, 1 ), LDC )
END IF
*
* W := W * V1**T
*
CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', N, K,
$ ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W**T
*
DO 30 J = 1, K
DO 20 I = 1, N
C( J, I ) = C( J, I ) - WORK( I, J )
20 CONTINUE
30 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H**T where C = ( C1 C2 )
*
* W := C * V = (C1*V1 + C2*V2) (stored in WORK)
*
* W := C1
*
DO 40 J = 1, K
CALL DCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 )
40 CONTINUE
*
* W := W * V1
*
CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit', M,
$ K, ONE, V, LDV, WORK, LDWORK )
IF( N.GT.K ) THEN
*
* W := W + C2 * V2
*
CALL DGEMM( 'No transpose', 'No transpose', M, K, N-K,
$ ONE, C( 1, K+1 ), LDC, V( K+1, 1 ), LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T**T
*
CALL DTRMM( 'Right', 'Upper', TRANS, 'Non-unit', M, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V**T
*
IF( N.GT.K ) THEN
*
* C2 := C2 - W * V2**T
*
CALL DGEMM( 'No transpose', 'Transpose', M, N-K, K,
$ -ONE, WORK, LDWORK, V( K+1, 1 ), LDV, ONE,
$ C( 1, K+1 ), LDC )
END IF
*
* W := W * V1**T
*
CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', M, K,
$ ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W
*
DO 60 J = 1, K
DO 50 I = 1, M
C( I, J ) = C( I, J ) - WORK( I, J )
50 CONTINUE
60 CONTINUE
END IF
*
ELSE
*
* Let V = ( V1 )
* ( V2 ) (last K rows)
* where V2 is unit upper triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H**T * C where C = ( C1 )
* ( C2 )
*
* W := C**T * V = (C1**T * V1 + C2**T * V2) (stored in WORK)
*
* W := C2**T
*
DO 70 J = 1, K
CALL DCOPY( N, C( M-K+J, 1 ), LDC, WORK( 1, J ), 1 )
70 CONTINUE
*
* W := W * V2
*
CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit', N,
$ K, ONE, V( M-K+1, 1 ), LDV, WORK, LDWORK )
IF( M.GT.K ) THEN
*
* W := W + C1**T * V1
*
CALL DGEMM( 'Transpose', 'No transpose', N, K, M-K,
$ ONE, C, LDC, V, LDV, ONE, WORK, LDWORK )
END IF
*
* W := W * T**T or W * T
*
CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - V * W**T
*
IF( M.GT.K ) THEN
*
* C1 := C1 - V1 * W**T
*
CALL DGEMM( 'No transpose', 'Transpose', M-K, N, K,
$ -ONE, V, LDV, WORK, LDWORK, ONE, C, LDC )
END IF
*
* W := W * V2**T
*
CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', N, K,
$ ONE, V( M-K+1, 1 ), LDV, WORK, LDWORK )
*
* C2 := C2 - W**T
*
DO 90 J = 1, K
DO 80 I = 1, N
C( M-K+J, I ) = C( M-K+J, I ) - WORK( I, J )
80 CONTINUE
90 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H**T where C = ( C1 C2 )
*
* W := C * V = (C1*V1 + C2*V2) (stored in WORK)
*
* W := C2
*
DO 100 J = 1, K
CALL DCOPY( M, C( 1, N-K+J ), 1, WORK( 1, J ), 1 )
100 CONTINUE
*
* W := W * V2
*
CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit', M,
$ K, ONE, V( N-K+1, 1 ), LDV, WORK, LDWORK )
IF( N.GT.K ) THEN
*
* W := W + C1 * V1
*
CALL DGEMM( 'No transpose', 'No transpose', M, K, N-K,
$ ONE, C, LDC, V, LDV, ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T**T
*
CALL DTRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V**T
*
IF( N.GT.K ) THEN
*
* C1 := C1 - W * V1**T
*
CALL DGEMM( 'No transpose', 'Transpose', M, N-K, K,
$ -ONE, WORK, LDWORK, V, LDV, ONE, C, LDC )
END IF
*
* W := W * V2**T
*
CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', M, K,
$ ONE, V( N-K+1, 1 ), LDV, WORK, LDWORK )
*
* C2 := C2 - W
*
DO 120 J = 1, K
DO 110 I = 1, M
C( I, N-K+J ) = C( I, N-K+J ) - WORK( I, J )
110 CONTINUE
120 CONTINUE
END IF
END IF
*
ELSE IF( LSAME( STOREV, 'R' ) ) THEN
*
IF( LSAME( DIRECT, 'F' ) ) THEN
*
* Let V = ( V1 V2 ) (V1: first K columns)
* where V1 is unit upper triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H**T * C where C = ( C1 )
* ( C2 )
*
* W := C**T * V**T = (C1**T * V1**T + C2**T * V2**T) (stored in WORK)
*
* W := C1**T
*
DO 130 J = 1, K
CALL DCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 )
130 CONTINUE
*
* W := W * V1**T
*
CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', N, K,
$ ONE, V, LDV, WORK, LDWORK )
IF( M.GT.K ) THEN
*
* W := W + C2**T * V2**T
*
CALL DGEMM( 'Transpose', 'Transpose', N, K, M-K, ONE,
$ C( K+1, 1 ), LDC, V( 1, K+1 ), LDV, ONE,
$ WORK, LDWORK )
END IF
*
* W := W * T**T or W * T
*
CALL DTRMM( 'Right', 'Upper', TRANST, 'Non-unit', N, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - V**T * W**T
*
IF( M.GT.K ) THEN
*
* C2 := C2 - V2**T * W**T
*
CALL DGEMM( 'Transpose', 'Transpose', M-K, N, K, -ONE,
$ V( 1, K+1 ), LDV, WORK, LDWORK, ONE,
$ C( K+1, 1 ), LDC )
END IF
*
* W := W * V1
*
CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit', N,
$ K, ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W**T
*
DO 150 J = 1, K
DO 140 I = 1, N
C( J, I ) = C( J, I ) - WORK( I, J )
140 CONTINUE
150 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H**T where C = ( C1 C2 )
*
* W := C * V**T = (C1*V1**T + C2*V2**T) (stored in WORK)
*
* W := C1
*
DO 160 J = 1, K
CALL DCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 )
160 CONTINUE
*
* W := W * V1**T
*
CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', M, K,
$ ONE, V, LDV, WORK, LDWORK )
IF( N.GT.K ) THEN
*
* W := W + C2 * V2**T
*
CALL DGEMM( 'No transpose', 'Transpose', M, K, N-K,
$ ONE, C( 1, K+1 ), LDC, V( 1, K+1 ), LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T**T
*
CALL DTRMM( 'Right', 'Upper', TRANS, 'Non-unit', M, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V
*
IF( N.GT.K ) THEN
*
* C2 := C2 - W * V2
*
CALL DGEMM( 'No transpose', 'No transpose', M, N-K, K,
$ -ONE, WORK, LDWORK, V( 1, K+1 ), LDV, ONE,
$ C( 1, K+1 ), LDC )
END IF
*
* W := W * V1
*
CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit', M,
$ K, ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W
*
DO 180 J = 1, K
DO 170 I = 1, M
C( I, J ) = C( I, J ) - WORK( I, J )
170 CONTINUE
180 CONTINUE
*
END IF
*
ELSE
*
* Let V = ( V1 V2 ) (V2: last K columns)
* where V2 is unit lower triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H**T * C where C = ( C1 )
* ( C2 )
*
* W := C**T * V**T = (C1**T * V1**T + C2**T * V2**T) (stored in WORK)
*
* W := C2**T
*
DO 190 J = 1, K
CALL DCOPY( N, C( M-K+J, 1 ), LDC, WORK( 1, J ), 1 )
190 CONTINUE
*
* W := W * V2**T
*
CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', N, K,
$ ONE, V( 1, M-K+1 ), LDV, WORK, LDWORK )
IF( M.GT.K ) THEN
*
* W := W + C1**T * V1**T
*
CALL DGEMM( 'Transpose', 'Transpose', N, K, M-K, ONE,
$ C, LDC, V, LDV, ONE, WORK, LDWORK )
END IF
*
* W := W * T**T or W * T
*
CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - V**T * W**T
*
IF( M.GT.K ) THEN
*
* C1 := C1 - V1**T * W**T
*
CALL DGEMM( 'Transpose', 'Transpose', M-K, N, K, -ONE,
$ V, LDV, WORK, LDWORK, ONE, C, LDC )
END IF
*
* W := W * V2
*
CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit', N,
$ K, ONE, V( 1, M-K+1 ), LDV, WORK, LDWORK )
*
* C2 := C2 - W**T
*
DO 210 J = 1, K
DO 200 I = 1, N
C( M-K+J, I ) = C( M-K+J, I ) - WORK( I, J )
200 CONTINUE
210 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H' where C = ( C1 C2 )
*
* W := C * V**T = (C1*V1**T + C2*V2**T) (stored in WORK)
*
* W := C2
*
DO 220 J = 1, K
CALL DCOPY( M, C( 1, N-K+J ), 1, WORK( 1, J ), 1 )
220 CONTINUE
*
* W := W * V2**T
*
CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', M, K,
$ ONE, V( 1, N-K+1 ), LDV, WORK, LDWORK )
IF( N.GT.K ) THEN
*
* W := W + C1 * V1**T
*
CALL DGEMM( 'No transpose', 'Transpose', M, K, N-K,
$ ONE, C, LDC, V, LDV, ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T**T
*
CALL DTRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V
*
IF( N.GT.K ) THEN
*
* C1 := C1 - W * V1
*
CALL DGEMM( 'No transpose', 'No transpose', M, N-K, K,
$ -ONE, WORK, LDWORK, V, LDV, ONE, C, LDC )
END IF
*
* W := W * V2
*
CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit', M,
$ K, ONE, V( 1, N-K+1 ), LDV, WORK, LDWORK )
*
* C1 := C1 - W
*
DO 240 J = 1, K
DO 230 I = 1, M
C( I, N-K+J ) = C( I, N-K+J ) - WORK( I, J )
230 CONTINUE
240 CONTINUE
*
END IF
*
END IF
END IF
*
RETURN
*
* End of DLARFB
*
END
*> \brief \b DLARFG generates an elementary reflector (Householder matrix).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFG + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
*
* .. Scalar Arguments ..
* INTEGER INCX, N
* DOUBLE PRECISION ALPHA, TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARFG generates a real elementary reflector H of order n, such
*> that
*>
*> H * ( alpha ) = ( beta ), H**T * H = I.
*> ( x ) ( 0 )
*>
*> where alpha and beta are scalars, and x is an (n-1)-element real
*> vector. H is represented in the form
*>
*> H = I - tau * ( 1 ) * ( 1 v**T ) ,
*> ( v )
*>
*> where tau is a real scalar and v is a real (n-1)-element
*> vector.
*>
*> If the elements of x are all zero, then tau = 0 and H is taken to be
*> the unit matrix.
*>
*> Otherwise 1 <= tau <= 2.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the elementary reflector.
*> \endverbatim
*>
*> \param[in,out] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> On entry, the value alpha.
*> On exit, it is overwritten with the value beta.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension
*> (1+(N-2)*abs(INCX))
*> On entry, the vector x.
*> On exit, it is overwritten with the vector v.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> The increment between elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The value tau.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INCX, N
DOUBLE PRECISION ALPHA, TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER J, KNT
DOUBLE PRECISION BETA, RSAFMN, SAFMIN, XNORM
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLAPY2, DNRM2
EXTERNAL DLAMCH, DLAPY2, DNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN
* ..
* .. External Subroutines ..
EXTERNAL DSCAL
* ..
* .. Executable Statements ..
*
IF( N.LE.1 ) THEN
TAU = ZERO
RETURN
END IF
*
XNORM = DNRM2( N-1, X, INCX )
*
IF( XNORM.EQ.ZERO ) THEN
*
* H = I
*
TAU = ZERO
ELSE
*
* general case
*
BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
SAFMIN = DLAMCH( 'S' ) / DLAMCH( 'E' )
KNT = 0
IF( ABS( BETA ).LT.SAFMIN ) THEN
*
* XNORM, BETA may be inaccurate; scale X and recompute them
*
RSAFMN = ONE / SAFMIN
10 CONTINUE
KNT = KNT + 1
CALL DSCAL( N-1, RSAFMN, X, INCX )
BETA = BETA*RSAFMN
ALPHA = ALPHA*RSAFMN
IF( ABS( BETA ).LT.SAFMIN )
$ GO TO 10
*
* New BETA is at most 1, at least SAFMIN
*
XNORM = DNRM2( N-1, X, INCX )
BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
END IF
TAU = ( BETA-ALPHA ) / BETA
CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
*
* If ALPHA is subnormal, it may lose relative accuracy
*
DO 20 J = 1, KNT
BETA = BETA*SAFMIN
20 CONTINUE
ALPHA = BETA
END IF
*
RETURN
*
* End of DLARFG
*
END
*> \brief \b DLARFT forms the triangular factor T of a block reflector H = I - vtvH
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFT + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, STOREV
* INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARFT forms the triangular factor T of a real block reflector H
*> of order n, which is defined as a product of k elementary reflectors.
*>
*> If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
*>
*> If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
*>
*> If STOREV = 'C', the vector which defines the elementary reflector
*> H(i) is stored in the i-th column of the array V, and
*>
*> H = I - V * T * V**T
*>
*> If STOREV = 'R', the vector which defines the elementary reflector
*> H(i) is stored in the i-th row of the array V, and
*>
*> H = I - V**T * T * V
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] DIRECT
*> \verbatim
*> DIRECT is CHARACTER*1
*> Specifies the order in which the elementary reflectors are
*> multiplied to form the block reflector:
*> = 'F': H = H(1) H(2) . . . H(k) (Forward)
*> = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*> STOREV is CHARACTER*1
*> Specifies how the vectors which define the elementary
*> reflectors are stored (see also Further Details):
*> = 'C': columnwise
*> = 'R': rowwise
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the block reflector H. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The order of the triangular factor T (= the number of
*> elementary reflectors). K >= 1.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (LDV,K) if STOREV = 'C'
*> (LDV,N) if STOREV = 'R'
*> The matrix V. See further details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V.
*> If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,K)
*> The k by k triangular factor T of the block reflector.
*> If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
*> lower triangular. The rest of the array is not used.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= K.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The shape of the matrix V and the storage of the vectors which define
*> the H(i) is best illustrated by the following example with n = 5 and
*> k = 3. The elements equal to 1 are not stored.
*>
*> DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
*>
*> V = ( 1 ) V = ( 1 v1 v1 v1 v1 )
*> ( v1 1 ) ( 1 v2 v2 v2 )
*> ( v1 v2 1 ) ( 1 v3 v3 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*>
*> DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
*>
*> V = ( v1 v2 v3 ) V = ( v1 v1 1 )
*> ( v1 v2 v3 ) ( v2 v2 v2 1 )
*> ( 1 v2 v3 ) ( v3 v3 v3 v3 1 )
*> ( 1 v3 )
*> ( 1 )
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER DIRECT, STOREV
INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, PREVLASTV, LASTV
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DTRMV
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( LSAME( DIRECT, 'F' ) ) THEN
PREVLASTV = N
DO I = 1, K
PREVLASTV = MAX( I, PREVLASTV )
IF( TAU( I ).EQ.ZERO ) THEN
*
* H(i) = I
*
DO J = 1, I
T( J, I ) = ZERO
END DO
ELSE
*
* general case
*
IF( LSAME( STOREV, 'C' ) ) THEN
* Skip any trailing zeros.
DO LASTV = N, I+1, -1
IF( V( LASTV, I ).NE.ZERO ) EXIT
END DO
DO J = 1, I-1
T( J, I ) = -TAU( I ) * V( I , J )
END DO
J = MIN( LASTV, PREVLASTV )
*
* T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)**T * V(i:j,i)
*
CALL DGEMV( 'Transpose', J-I, I-1, -TAU( I ),
$ V( I+1, 1 ), LDV, V( I+1, I ), 1, ONE,
$ T( 1, I ), 1 )
ELSE
* Skip any trailing zeros.
DO LASTV = N, I+1, -1
IF( V( I, LASTV ).NE.ZERO ) EXIT
END DO
DO J = 1, I-1
T( J, I ) = -TAU( I ) * V( J , I )
END DO
J = MIN( LASTV, PREVLASTV )
*
* T(1:i-1,i) := - tau(i) * V(1:i-1,i:j) * V(i,i:j)**T
*
CALL DGEMV( 'No transpose', I-1, J-I, -TAU( I ),
$ V( 1, I+1 ), LDV, V( I, I+1 ), LDV, ONE,
$ T( 1, I ), 1 )
END IF
*
* T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
*
CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T,
$ LDT, T( 1, I ), 1 )
T( I, I ) = TAU( I )
IF( I.GT.1 ) THEN
PREVLASTV = MAX( PREVLASTV, LASTV )
ELSE
PREVLASTV = LASTV
END IF
END IF
END DO
ELSE
PREVLASTV = 1
DO I = K, 1, -1
IF( TAU( I ).EQ.ZERO ) THEN
*
* H(i) = I
*
DO J = I, K
T( J, I ) = ZERO
END DO
ELSE
*
* general case
*
IF( I.LT.K ) THEN
IF( LSAME( STOREV, 'C' ) ) THEN
* Skip any leading zeros.
DO LASTV = 1, I-1
IF( V( LASTV, I ).NE.ZERO ) EXIT
END DO
DO J = I+1, K
T( J, I ) = -TAU( I ) * V( N-K+I , J )
END DO
J = MAX( LASTV, PREVLASTV )
*
* T(i+1:k,i) = -tau(i) * V(j:n-k+i,i+1:k)**T * V(j:n-k+i,i)
*
CALL DGEMV( 'Transpose', N-K+I-J, K-I, -TAU( I ),
$ V( J, I+1 ), LDV, V( J, I ), 1, ONE,
$ T( I+1, I ), 1 )
ELSE
* Skip any leading zeros.
DO LASTV = 1, I-1
IF( V( I, LASTV ).NE.ZERO ) EXIT
END DO
DO J = I+1, K
T( J, I ) = -TAU( I ) * V( J, N-K+I )
END DO
J = MAX( LASTV, PREVLASTV )
*
* T(i+1:k,i) = -tau(i) * V(i+1:k,j:n-k+i) * V(i,j:n-k+i)**T
*
CALL DGEMV( 'No transpose', K-I, N-K+I-J,
$ -TAU( I ), V( I+1, J ), LDV, V( I, J ), LDV,
$ ONE, T( I+1, I ), 1 )
END IF
*
* T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i)
*
CALL DTRMV( 'Lower', 'No transpose', 'Non-unit', K-I,
$ T( I+1, I+1 ), LDT, T( I+1, I ), 1 )
IF( I.GT.1 ) THEN
PREVLASTV = MIN( PREVLASTV, LASTV )
ELSE
PREVLASTV = LASTV
END IF
END IF
T( I, I ) = TAU( I )
END IF
END DO
END IF
RETURN
*
* End of DLARFT
*
END
*> \brief \b DLARFX applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFX + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARFX( SIDE, M, N, V, TAU, C, LDC, WORK )
*
* .. Scalar Arguments ..
* CHARACTER SIDE
* INTEGER LDC, M, N
* DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARFX applies a real elementary reflector H to a real m by n
*> matrix C, from either the left or the right. H is represented in the
*> form
*>
*> H = I - tau * v * v**T
*>
*> where tau is a real scalar and v is a real vector.
*>
*> If tau = 0, then H is taken to be the unit matrix
*>
*> This version uses inline code if H has order < 11.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': form H * C
*> = 'R': form C * H
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (M) if SIDE = 'L'
*> or (N) if SIDE = 'R'
*> The vector v in the representation of H.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The value tau in the representation of H.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by the matrix H * C if SIDE = 'L',
*> or C * H if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDA >= (1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L'
*> or (M) if SIDE = 'R'
*> WORK is not referenced if H has order < 11.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLARFX( SIDE, M, N, V, TAU, C, LDC, WORK )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER SIDE
INTEGER LDC, M, N
DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER J
DOUBLE PRECISION SUM, T1, T10, T2, T3, T4, T5, T6, T7, T8, T9,
$ V1, V10, V2, V3, V4, V5, V6, V7, V8, V9
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLARF
* ..
* .. Executable Statements ..
*
IF( TAU.EQ.ZERO )
$ RETURN
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C, where H has order m.
*
GO TO ( 10, 30, 50, 70, 90, 110, 130, 150,
$ 170, 190 )M
*
* Code for general M
*
CALL DLARF( SIDE, M, N, V, 1, TAU, C, LDC, WORK )
GO TO 410
10 CONTINUE
*
* Special code for 1 x 1 Householder
*
T1 = ONE - TAU*V( 1 )*V( 1 )
DO 20 J = 1, N
C( 1, J ) = T1*C( 1, J )
20 CONTINUE
GO TO 410
30 CONTINUE
*
* Special code for 2 x 2 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
DO 40 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
40 CONTINUE
GO TO 410
50 CONTINUE
*
* Special code for 3 x 3 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
DO 60 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
60 CONTINUE
GO TO 410
70 CONTINUE
*
* Special code for 4 x 4 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
DO 80 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
$ V4*C( 4, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
C( 4, J ) = C( 4, J ) - SUM*T4
80 CONTINUE
GO TO 410
90 CONTINUE
*
* Special code for 5 x 5 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
DO 100 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
$ V4*C( 4, J ) + V5*C( 5, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
C( 4, J ) = C( 4, J ) - SUM*T4
C( 5, J ) = C( 5, J ) - SUM*T5
100 CONTINUE
GO TO 410
110 CONTINUE
*
* Special code for 6 x 6 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
DO 120 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
$ V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
C( 4, J ) = C( 4, J ) - SUM*T4
C( 5, J ) = C( 5, J ) - SUM*T5
C( 6, J ) = C( 6, J ) - SUM*T6
120 CONTINUE
GO TO 410
130 CONTINUE
*
* Special code for 7 x 7 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
DO 140 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
$ V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) +
$ V7*C( 7, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
C( 4, J ) = C( 4, J ) - SUM*T4
C( 5, J ) = C( 5, J ) - SUM*T5
C( 6, J ) = C( 6, J ) - SUM*T6
C( 7, J ) = C( 7, J ) - SUM*T7
140 CONTINUE
GO TO 410
150 CONTINUE
*
* Special code for 8 x 8 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
V8 = V( 8 )
T8 = TAU*V8
DO 160 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
$ V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) +
$ V7*C( 7, J ) + V8*C( 8, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
C( 4, J ) = C( 4, J ) - SUM*T4
C( 5, J ) = C( 5, J ) - SUM*T5
C( 6, J ) = C( 6, J ) - SUM*T6
C( 7, J ) = C( 7, J ) - SUM*T7
C( 8, J ) = C( 8, J ) - SUM*T8
160 CONTINUE
GO TO 410
170 CONTINUE
*
* Special code for 9 x 9 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
V8 = V( 8 )
T8 = TAU*V8
V9 = V( 9 )
T9 = TAU*V9
DO 180 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
$ V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) +
$ V7*C( 7, J ) + V8*C( 8, J ) + V9*C( 9, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
C( 4, J ) = C( 4, J ) - SUM*T4
C( 5, J ) = C( 5, J ) - SUM*T5
C( 6, J ) = C( 6, J ) - SUM*T6
C( 7, J ) = C( 7, J ) - SUM*T7
C( 8, J ) = C( 8, J ) - SUM*T8
C( 9, J ) = C( 9, J ) - SUM*T9
180 CONTINUE
GO TO 410
190 CONTINUE
*
* Special code for 10 x 10 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
V8 = V( 8 )
T8 = TAU*V8
V9 = V( 9 )
T9 = TAU*V9
V10 = V( 10 )
T10 = TAU*V10
DO 200 J = 1, N
SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
$ V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) +
$ V7*C( 7, J ) + V8*C( 8, J ) + V9*C( 9, J ) +
$ V10*C( 10, J )
C( 1, J ) = C( 1, J ) - SUM*T1
C( 2, J ) = C( 2, J ) - SUM*T2
C( 3, J ) = C( 3, J ) - SUM*T3
C( 4, J ) = C( 4, J ) - SUM*T4
C( 5, J ) = C( 5, J ) - SUM*T5
C( 6, J ) = C( 6, J ) - SUM*T6
C( 7, J ) = C( 7, J ) - SUM*T7
C( 8, J ) = C( 8, J ) - SUM*T8
C( 9, J ) = C( 9, J ) - SUM*T9
C( 10, J ) = C( 10, J ) - SUM*T10
200 CONTINUE
GO TO 410
ELSE
*
* Form C * H, where H has order n.
*
GO TO ( 210, 230, 250, 270, 290, 310, 330, 350,
$ 370, 390 )N
*
* Code for general N
*
CALL DLARF( SIDE, M, N, V, 1, TAU, C, LDC, WORK )
GO TO 410
210 CONTINUE
*
* Special code for 1 x 1 Householder
*
T1 = ONE - TAU*V( 1 )*V( 1 )
DO 220 J = 1, M
C( J, 1 ) = T1*C( J, 1 )
220 CONTINUE
GO TO 410
230 CONTINUE
*
* Special code for 2 x 2 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
DO 240 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
240 CONTINUE
GO TO 410
250 CONTINUE
*
* Special code for 3 x 3 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
DO 260 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
260 CONTINUE
GO TO 410
270 CONTINUE
*
* Special code for 4 x 4 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
DO 280 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
$ V4*C( J, 4 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
C( J, 4 ) = C( J, 4 ) - SUM*T4
280 CONTINUE
GO TO 410
290 CONTINUE
*
* Special code for 5 x 5 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
DO 300 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
$ V4*C( J, 4 ) + V5*C( J, 5 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
C( J, 4 ) = C( J, 4 ) - SUM*T4
C( J, 5 ) = C( J, 5 ) - SUM*T5
300 CONTINUE
GO TO 410
310 CONTINUE
*
* Special code for 6 x 6 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
DO 320 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
$ V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
C( J, 4 ) = C( J, 4 ) - SUM*T4
C( J, 5 ) = C( J, 5 ) - SUM*T5
C( J, 6 ) = C( J, 6 ) - SUM*T6
320 CONTINUE
GO TO 410
330 CONTINUE
*
* Special code for 7 x 7 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
DO 340 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
$ V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) +
$ V7*C( J, 7 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
C( J, 4 ) = C( J, 4 ) - SUM*T4
C( J, 5 ) = C( J, 5 ) - SUM*T5
C( J, 6 ) = C( J, 6 ) - SUM*T6
C( J, 7 ) = C( J, 7 ) - SUM*T7
340 CONTINUE
GO TO 410
350 CONTINUE
*
* Special code for 8 x 8 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
V8 = V( 8 )
T8 = TAU*V8
DO 360 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
$ V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) +
$ V7*C( J, 7 ) + V8*C( J, 8 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
C( J, 4 ) = C( J, 4 ) - SUM*T4
C( J, 5 ) = C( J, 5 ) - SUM*T5
C( J, 6 ) = C( J, 6 ) - SUM*T6
C( J, 7 ) = C( J, 7 ) - SUM*T7
C( J, 8 ) = C( J, 8 ) - SUM*T8
360 CONTINUE
GO TO 410
370 CONTINUE
*
* Special code for 9 x 9 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
V8 = V( 8 )
T8 = TAU*V8
V9 = V( 9 )
T9 = TAU*V9
DO 380 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
$ V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) +
$ V7*C( J, 7 ) + V8*C( J, 8 ) + V9*C( J, 9 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
C( J, 4 ) = C( J, 4 ) - SUM*T4
C( J, 5 ) = C( J, 5 ) - SUM*T5
C( J, 6 ) = C( J, 6 ) - SUM*T6
C( J, 7 ) = C( J, 7 ) - SUM*T7
C( J, 8 ) = C( J, 8 ) - SUM*T8
C( J, 9 ) = C( J, 9 ) - SUM*T9
380 CONTINUE
GO TO 410
390 CONTINUE
*
* Special code for 10 x 10 Householder
*
V1 = V( 1 )
T1 = TAU*V1
V2 = V( 2 )
T2 = TAU*V2
V3 = V( 3 )
T3 = TAU*V3
V4 = V( 4 )
T4 = TAU*V4
V5 = V( 5 )
T5 = TAU*V5
V6 = V( 6 )
T6 = TAU*V6
V7 = V( 7 )
T7 = TAU*V7
V8 = V( 8 )
T8 = TAU*V8
V9 = V( 9 )
T9 = TAU*V9
V10 = V( 10 )
T10 = TAU*V10
DO 400 J = 1, M
SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
$ V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) +
$ V7*C( J, 7 ) + V8*C( J, 8 ) + V9*C( J, 9 ) +
$ V10*C( J, 10 )
C( J, 1 ) = C( J, 1 ) - SUM*T1
C( J, 2 ) = C( J, 2 ) - SUM*T2
C( J, 3 ) = C( J, 3 ) - SUM*T3
C( J, 4 ) = C( J, 4 ) - SUM*T4
C( J, 5 ) = C( J, 5 ) - SUM*T5
C( J, 6 ) = C( J, 6 ) - SUM*T6
C( J, 7 ) = C( J, 7 ) - SUM*T7
C( J, 8 ) = C( J, 8 ) - SUM*T8
C( J, 9 ) = C( J, 9 ) - SUM*T9
C( J, 10 ) = C( J, 10 ) - SUM*T10
400 CONTINUE
GO TO 410
END IF
410 CONTINUE
RETURN
*
* End of DLARFX
*
END
*> \brief \b DLARTG generates a plane rotation with real cosine and real sine.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARTG + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARTG( F, G, CS, SN, R )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION CS, F, G, R, SN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARTG generate a plane rotation so that
*>
*> [ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1.
*> [ -SN CS ] [ G ] [ 0 ]
*>
*> This is a slower, more accurate version of the BLAS1 routine DROTG,
*> with the following other differences:
*> F and G are unchanged on return.
*> If G=0, then CS=1 and SN=0.
*> If F=0 and (G .ne. 0), then CS=0 and SN=1 without doing any
*> floating point operations (saves work in DBDSQR when
*> there are zeros on the diagonal).
*>
*> If F exceeds G in magnitude, CS will be positive.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] F
*> \verbatim
*> F is DOUBLE PRECISION
*> The first component of vector to be rotated.
*> \endverbatim
*>
*> \param[in] G
*> \verbatim
*> G is DOUBLE PRECISION
*> The second component of vector to be rotated.
*> \endverbatim
*>
*> \param[out] CS
*> \verbatim
*> CS is DOUBLE PRECISION
*> The cosine of the rotation.
*> \endverbatim
*>
*> \param[out] SN
*> \verbatim
*> SN is DOUBLE PRECISION
*> The sine of the rotation.
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*> R is DOUBLE PRECISION
*> The nonzero component of the rotated vector.
*>
*> This version has a few statements commented out for thread safety
*> (machine parameters are computed on each entry). 10 feb 03, SJH.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLARTG( F, G, CS, SN, R )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
DOUBLE PRECISION CS, F, G, R, SN
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
* ..
* .. Local Scalars ..
* LOGICAL FIRST
INTEGER COUNT, I
DOUBLE PRECISION EPS, F1, G1, SAFMIN, SAFMN2, SAFMX2, SCALE
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, LOG, MAX, SQRT
* ..
* .. Save statement ..
* SAVE FIRST, SAFMX2, SAFMIN, SAFMN2
* ..
* .. Data statements ..
* DATA FIRST / .TRUE. /
* ..
* .. Executable Statements ..
*
* IF( FIRST ) THEN
SAFMIN = DLAMCH( 'S' )
EPS = DLAMCH( 'E' )
SAFMN2 = DLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) /
$ LOG( DLAMCH( 'B' ) ) / TWO )
SAFMX2 = ONE / SAFMN2
* FIRST = .FALSE.
* END IF
IF( G.EQ.ZERO ) THEN
CS = ONE
SN = ZERO
R = F
ELSE IF( F.EQ.ZERO ) THEN
CS = ZERO
SN = ONE
R = G
ELSE
F1 = F
G1 = G
SCALE = MAX( ABS( F1 ), ABS( G1 ) )
IF( SCALE.GE.SAFMX2 ) THEN
COUNT = 0
10 CONTINUE
COUNT = COUNT + 1
F1 = F1*SAFMN2
G1 = G1*SAFMN2
SCALE = MAX( ABS( F1 ), ABS( G1 ) )
IF( SCALE.GE.SAFMX2 )
$ GO TO 10
R = SQRT( F1**2+G1**2 )
CS = F1 / R
SN = G1 / R
DO 20 I = 1, COUNT
R = R*SAFMX2
20 CONTINUE
ELSE IF( SCALE.LE.SAFMN2 ) THEN
COUNT = 0
30 CONTINUE
COUNT = COUNT + 1
F1 = F1*SAFMX2
G1 = G1*SAFMX2
SCALE = MAX( ABS( F1 ), ABS( G1 ) )
IF( SCALE.LE.SAFMN2 )
$ GO TO 30
R = SQRT( F1**2+G1**2 )
CS = F1 / R
SN = G1 / R
DO 40 I = 1, COUNT
R = R*SAFMN2
40 CONTINUE
ELSE
R = SQRT( F1**2+G1**2 )
CS = F1 / R
SN = G1 / R
END IF
IF( ABS( F ).GT.ABS( G ) .AND. CS.LT.ZERO ) THEN
CS = -CS
SN = -SN
R = -R
END IF
END IF
RETURN
*
* End of DLARTG
*
END
*> \brief \b DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASCL + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASCL( TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TYPE
* INTEGER INFO, KL, KU, LDA, M, N
* DOUBLE PRECISION CFROM, CTO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASCL multiplies the M by N real matrix A by the real scalar
*> CTO/CFROM. This is done without over/underflow as long as the final
*> result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that
*> A may be full, upper triangular, lower triangular, upper Hessenberg,
*> or banded.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TYPE
*> \verbatim
*> TYPE is CHARACTER*1
*> TYPE indices the storage type of the input matrix.
*> = 'G': A is a full matrix.
*> = 'L': A is a lower triangular matrix.
*> = 'U': A is an upper triangular matrix.
*> = 'H': A is an upper Hessenberg matrix.
*> = 'B': A is a symmetric band matrix with lower bandwidth KL
*> and upper bandwidth KU and with the only the lower
*> half stored.
*> = 'Q': A is a symmetric band matrix with lower bandwidth KL
*> and upper bandwidth KU and with the only the upper
*> half stored.
*> = 'Z': A is a band matrix with lower bandwidth KL and upper
*> bandwidth KU. See DGBTRF for storage details.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The lower bandwidth of A. Referenced only if TYPE = 'B',
*> 'Q' or 'Z'.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The upper bandwidth of A. Referenced only if TYPE = 'B',
*> 'Q' or 'Z'.
*> \endverbatim
*>
*> \param[in] CFROM
*> \verbatim
*> CFROM is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] CTO
*> \verbatim
*> CTO is DOUBLE PRECISION
*>
*> The matrix A is multiplied by CTO/CFROM. A(I,J) is computed
*> without over/underflow if the final result CTO*A(I,J)/CFROM
*> can be represented without over/underflow. CFROM must be
*> nonzero.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The matrix to be multiplied by CTO/CFROM. See TYPE for the
*> storage type.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If TYPE = 'G', 'L', 'U', 'H', LDA >= max(1,M);
*> TYPE = 'B', LDA >= KL+1;
*> TYPE = 'Q', LDA >= KU+1;
*> TYPE = 'Z', LDA >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> 0 - successful exit
*> <0 - if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLASCL( TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO )
*
* -- LAPACK auxiliary routine (version 3.6.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
CHARACTER TYPE
INTEGER INFO, KL, KU, LDA, M, N
DOUBLE PRECISION CFROM, CTO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL DONE
INTEGER I, ITYPE, J, K1, K2, K3, K4
DOUBLE PRECISION BIGNUM, CFROM1, CFROMC, CTO1, CTOC, MUL, SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
*
IF( LSAME( TYPE, 'G' ) ) THEN
ITYPE = 0
ELSE IF( LSAME( TYPE, 'L' ) ) THEN
ITYPE = 1
ELSE IF( LSAME( TYPE, 'U' ) ) THEN
ITYPE = 2
ELSE IF( LSAME( TYPE, 'H' ) ) THEN
ITYPE = 3
ELSE IF( LSAME( TYPE, 'B' ) ) THEN
ITYPE = 4
ELSE IF( LSAME( TYPE, 'Q' ) ) THEN
ITYPE = 5
ELSE IF( LSAME( TYPE, 'Z' ) ) THEN
ITYPE = 6
ELSE
ITYPE = -1
END IF
*
IF( ITYPE.EQ.-1 ) THEN
INFO = -1
ELSE IF( CFROM.EQ.ZERO .OR. DISNAN(CFROM) ) THEN
INFO = -4
ELSE IF( DISNAN(CTO) ) THEN
INFO = -5
ELSE IF( M.LT.0 ) THEN
INFO = -6
ELSE IF( N.LT.0 .OR. ( ITYPE.EQ.4 .AND. N.NE.M ) .OR.
$ ( ITYPE.EQ.5 .AND. N.NE.M ) ) THEN
INFO = -7
ELSE IF( ITYPE.LE.3 .AND. LDA.LT.MAX( 1, M ) ) THEN
INFO = -9
ELSE IF( ITYPE.GE.4 ) THEN
IF( KL.LT.0 .OR. KL.GT.MAX( M-1, 0 ) ) THEN
INFO = -2
ELSE IF( KU.LT.0 .OR. KU.GT.MAX( N-1, 0 ) .OR.
$ ( ( ITYPE.EQ.4 .OR. ITYPE.EQ.5 ) .AND. KL.NE.KU ) )
$ THEN
INFO = -3
ELSE IF( ( ITYPE.EQ.4 .AND. LDA.LT.KL+1 ) .OR.
$ ( ITYPE.EQ.5 .AND. LDA.LT.KU+1 ) .OR.
$ ( ITYPE.EQ.6 .AND. LDA.LT.2*KL+KU+1 ) ) THEN
INFO = -9
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASCL', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. M.EQ.0 )
$ RETURN
*
* Get machine parameters
*
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
*
CFROMC = CFROM
CTOC = CTO
*
10 CONTINUE
CFROM1 = CFROMC*SMLNUM
IF( CFROM1.EQ.CFROMC ) THEN
! CFROMC is an inf. Multiply by a correctly signed zero for
! finite CTOC, or a NaN if CTOC is infinite.
MUL = CTOC / CFROMC
DONE = .TRUE.
CTO1 = CTOC
ELSE
CTO1 = CTOC / BIGNUM
IF( CTO1.EQ.CTOC ) THEN
! CTOC is either 0 or an inf. In both cases, CTOC itself
! serves as the correct multiplication factor.
MUL = CTOC
DONE = .TRUE.
CFROMC = ONE
ELSE IF( ABS( CFROM1 ).GT.ABS( CTOC ) .AND. CTOC.NE.ZERO ) THEN
MUL = SMLNUM
DONE = .FALSE.
CFROMC = CFROM1
ELSE IF( ABS( CTO1 ).GT.ABS( CFROMC ) ) THEN
MUL = BIGNUM
DONE = .FALSE.
CTOC = CTO1
ELSE
MUL = CTOC / CFROMC
DONE = .TRUE.
END IF
END IF
*
IF( ITYPE.EQ.0 ) THEN
*
* Full matrix
*
DO 30 J = 1, N
DO 20 I = 1, M
A( I, J ) = A( I, J )*MUL
20 CONTINUE
30 CONTINUE
*
ELSE IF( ITYPE.EQ.1 ) THEN
*
* Lower triangular matrix
*
DO 50 J = 1, N
DO 40 I = J, M
A( I, J ) = A( I, J )*MUL
40 CONTINUE
50 CONTINUE
*
ELSE IF( ITYPE.EQ.2 ) THEN
*
* Upper triangular matrix
*
DO 70 J = 1, N
DO 60 I = 1, MIN( J, M )
A( I, J ) = A( I, J )*MUL
60 CONTINUE
70 CONTINUE
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
* Upper Hessenberg matrix
*
DO 90 J = 1, N
DO 80 I = 1, MIN( J+1, M )
A( I, J ) = A( I, J )*MUL
80 CONTINUE
90 CONTINUE
*
ELSE IF( ITYPE.EQ.4 ) THEN
*
* Lower half of a symmetric band matrix
*
K3 = KL + 1
K4 = N + 1
DO 110 J = 1, N
DO 100 I = 1, MIN( K3, K4-J )
A( I, J ) = A( I, J )*MUL
100 CONTINUE
110 CONTINUE
*
ELSE IF( ITYPE.EQ.5 ) THEN
*
* Upper half of a symmetric band matrix
*
K1 = KU + 2
K3 = KU + 1
DO 130 J = 1, N
DO 120 I = MAX( K1-J, 1 ), K3
A( I, J ) = A( I, J )*MUL
120 CONTINUE
130 CONTINUE
*
ELSE IF( ITYPE.EQ.6 ) THEN
*
* Band matrix
*
K1 = KL + KU + 2
K2 = KL + 1
K3 = 2*KL + KU + 1
K4 = KL + KU + 1 + M
DO 150 J = 1, N
DO 140 I = MAX( K1-J, K2 ), MIN( K3, K4-J )
A( I, J ) = A( I, J )*MUL
140 CONTINUE
150 CONTINUE
*
END IF
*
IF( .NOT.DONE )
$ GO TO 10
*
RETURN
*
* End of DLASCL
*
END
*> \brief \b DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASET + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASET( UPLO, M, N, ALPHA, BETA, A, LDA )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDA, M, N
* DOUBLE PRECISION ALPHA, BETA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASET initializes an m-by-n matrix A to BETA on the diagonal and
*> ALPHA on the offdiagonals.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies the part of the matrix A to be set.
*> = 'U': Upper triangular part is set; the strictly lower
*> triangular part of A is not changed.
*> = 'L': Lower triangular part is set; the strictly upper
*> triangular part of A is not changed.
*> Otherwise: All of the matrix A is set.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> The constant to which the offdiagonal elements are to be set.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION
*> The constant to which the diagonal elements are to be set.
*> \endverbatim
*>
*> \param[out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On exit, the leading m-by-n submatrix of A is set as follows:
*>
*> if UPLO = 'U', A(i,j) = ALPHA, 1<=i<=j-1, 1<=j<=n,
*> if UPLO = 'L', A(i,j) = ALPHA, j+1<=i<=m, 1<=j<=n,
*> otherwise, A(i,j) = ALPHA, 1<=i<=m, 1<=j<=n, i.ne.j,
*>
*> and, for all UPLO, A(i,i) = BETA, 1<=i<=min(m,n).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2015
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLASET( UPLO, M, N, ALPHA, BETA, A, LDA )
*
* -- LAPACK auxiliary routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2015
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, M, N
DOUBLE PRECISION ALPHA, BETA
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Set the strictly upper triangular or trapezoidal part of the
* array to ALPHA.
*
DO 20 J = 2, N
DO 10 I = 1, MIN( J-1, M )
A( I, J ) = ALPHA
10 CONTINUE
20 CONTINUE
*
ELSE IF( LSAME( UPLO, 'L' ) ) THEN
*
* Set the strictly lower triangular or trapezoidal part of the
* array to ALPHA.
*
DO 40 J = 1, MIN( M, N )
DO 30 I = J + 1, M
A( I, J ) = ALPHA
30 CONTINUE
40 CONTINUE
*
ELSE
*
* Set the leading m-by-n submatrix to ALPHA.
*
DO 60 J = 1, N
DO 50 I = 1, M
A( I, J ) = ALPHA
50 CONTINUE
60 CONTINUE
END IF
*
* Set the first min(M,N) diagonal elements to BETA.
*
DO 70 I = 1, MIN( M, N )
A( I, I ) = BETA
70 CONTINUE
*
RETURN
*
* End of DLASET
*
END
*> \brief \b DLASSQ updates a sum of squares represented in scaled form.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASSQ + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASSQ( N, X, INCX, SCALE, SUMSQ )
*
* .. Scalar Arguments ..
* INTEGER INCX, N
* DOUBLE PRECISION SCALE, SUMSQ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASSQ returns the values scl and smsq such that
*>
*> ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq,
*>
*> where x( i ) = X( 1 + ( i - 1 )*INCX ). The value of sumsq is
*> assumed to be non-negative and scl returns the value
*>
*> scl = max( scale, abs( x( i ) ) ).
*>
*> scale and sumsq must be supplied in SCALE and SUMSQ and
*> scl and smsq are overwritten on SCALE and SUMSQ respectively.
*>
*> The routine makes only one pass through the vector x.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of elements to be used from the vector X.
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (N)
*> The vector for which a scaled sum of squares is computed.
*> x( i ) = X( 1 + ( i - 1 )*INCX ), 1 <= i <= n.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> The increment between successive values of the vector X.
*> INCX > 0.
*> \endverbatim
*>
*> \param[in,out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> On entry, the value scale in the equation above.
*> On exit, SCALE is overwritten with scl , the scaling factor
*> for the sum of squares.
*> \endverbatim
*>
*> \param[in,out] SUMSQ
*> \verbatim
*> SUMSQ is DOUBLE PRECISION
*> On entry, the value sumsq in the equation above.
*> On exit, SUMSQ is overwritten with smsq , the basic sum of
*> squares from which scl has been factored out.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLASSQ( N, X, INCX, SCALE, SUMSQ )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INCX, N
DOUBLE PRECISION SCALE, SUMSQ
* ..
* .. Array Arguments ..
DOUBLE PRECISION X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER IX
DOUBLE PRECISION ABSXI
* ..
* .. External Functions ..
LOGICAL DISNAN
EXTERNAL DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
IF( N.GT.0 ) THEN
DO 10 IX = 1, 1 + ( N-1 )*INCX, INCX
ABSXI = ABS( X( IX ) )
IF( ABSXI.GT.ZERO.OR.DISNAN( ABSXI ) ) THEN
IF( SCALE.LT.ABSXI ) THEN
SUMSQ = 1 + SUMSQ*( SCALE / ABSXI )**2
SCALE = ABSXI
ELSE
SUMSQ = SUMSQ + ( ABSXI / SCALE )**2
END IF
END IF
10 CONTINUE
END IF
RETURN
*
* End of DLASSQ
*
END
*> \brief \b DLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASY2 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
* LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
*
* .. Scalar Arguments ..
* LOGICAL LTRANL, LTRANR
* INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
* DOUBLE PRECISION SCALE, XNORM
* ..
* .. Array Arguments ..
* DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in
*>
*> op(TL)*X + ISGN*X*op(TR) = SCALE*B,
*>
*> where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or
*> -1. op(T) = T or T**T, where T**T denotes the transpose of T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] LTRANL
*> \verbatim
*> LTRANL is LOGICAL
*> On entry, LTRANL specifies the op(TL):
*> = .FALSE., op(TL) = TL,
*> = .TRUE., op(TL) = TL**T.
*> \endverbatim
*>
*> \param[in] LTRANR
*> \verbatim
*> LTRANR is LOGICAL
*> On entry, LTRANR specifies the op(TR):
*> = .FALSE., op(TR) = TR,
*> = .TRUE., op(TR) = TR**T.
*> \endverbatim
*>
*> \param[in] ISGN
*> \verbatim
*> ISGN is INTEGER
*> On entry, ISGN specifies the sign of the equation
*> as described before. ISGN may only be 1 or -1.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> On entry, N1 specifies the order of matrix TL.
*> N1 may only be 0, 1 or 2.
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*> N2 is INTEGER
*> On entry, N2 specifies the order of matrix TR.
*> N2 may only be 0, 1 or 2.
*> \endverbatim
*>
*> \param[in] TL
*> \verbatim
*> TL is DOUBLE PRECISION array, dimension (LDTL,2)
*> On entry, TL contains an N1 by N1 matrix.
*> \endverbatim
*>
*> \param[in] LDTL
*> \verbatim
*> LDTL is INTEGER
*> The leading dimension of the matrix TL. LDTL >= max(1,N1).
*> \endverbatim
*>
*> \param[in] TR
*> \verbatim
*> TR is DOUBLE PRECISION array, dimension (LDTR,2)
*> On entry, TR contains an N2 by N2 matrix.
*> \endverbatim
*>
*> \param[in] LDTR
*> \verbatim
*> LDTR is INTEGER
*> The leading dimension of the matrix TR. LDTR >= max(1,N2).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,2)
*> On entry, the N1 by N2 matrix B contains the right-hand
*> side of the equation.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the matrix B. LDB >= max(1,N1).
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> On exit, SCALE contains the scale factor. SCALE is chosen
*> less than or equal to 1 to prevent the solution overflowing.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,2)
*> On exit, X contains the N1 by N2 solution.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the matrix X. LDX >= max(1,N1).
*> \endverbatim
*>
*> \param[out] XNORM
*> \verbatim
*> XNORM is DOUBLE PRECISION
*> On exit, XNORM is the infinity-norm of the solution.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> On exit, INFO is set to
*> 0: successful exit.
*> 1: TL and TR have too close eigenvalues, so TL or
*> TR is perturbed to get a nonsingular equation.
*> NOTE: In the interests of speed, this routine does not
*> check the inputs for errors.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup doubleSYauxiliary
*
* =====================================================================
SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
$ LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
*
* -- LAPACK auxiliary routine (version 3.6.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
LOGICAL LTRANL, LTRANR
INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
DOUBLE PRECISION SCALE, XNORM
* ..
* .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION TWO, HALF, EIGHT
PARAMETER ( TWO = 2.0D+0, HALF = 0.5D+0, EIGHT = 8.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL BSWAP, XSWAP
INTEGER I, IP, IPIV, IPSV, J, JP, JPSV, K
DOUBLE PRECISION BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1,
$ TEMP, U11, U12, U22, XMAX
* ..
* .. Local Arrays ..
LOGICAL BSWPIV( 4 ), XSWPIV( 4 )
INTEGER JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ),
$ LOCU22( 4 )
DOUBLE PRECISION BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 )
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL IDAMAX, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Data statements ..
DATA LOCU12 / 3, 4, 1, 2 / , LOCL21 / 2, 1, 4, 3 / ,
$ LOCU22 / 4, 3, 2, 1 /
DATA XSWPIV / .FALSE., .FALSE., .TRUE., .TRUE. /
DATA BSWPIV / .FALSE., .TRUE., .FALSE., .TRUE. /
* ..
* .. Executable Statements ..
*
* Do not check the input parameters for errors
*
INFO = 0
*
* Quick return if possible
*
IF( N1.EQ.0 .OR. N2.EQ.0 )
$ RETURN
*
* Set constants to control overflow
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
SGN = ISGN
*
K = N1 + N1 + N2 - 2
GO TO ( 10, 20, 30, 50 )K
*
* 1 by 1: TL11*X + SGN*X*TR11 = B11
*
10 CONTINUE
TAU1 = TL( 1, 1 ) + SGN*TR( 1, 1 )
BET = ABS( TAU1 )
IF( BET.LE.SMLNUM ) THEN
TAU1 = SMLNUM
BET = SMLNUM
INFO = 1
END IF
*
SCALE = ONE
GAM = ABS( B( 1, 1 ) )
IF( SMLNUM*GAM.GT.BET )
$ SCALE = ONE / GAM
*
X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / TAU1
XNORM = ABS( X( 1, 1 ) )
RETURN
*
* 1 by 2:
* TL11*[X11 X12] + ISGN*[X11 X12]*op[TR11 TR12] = [B11 B12]
* [TR21 TR22]
*
20 CONTINUE
*
SMIN = MAX( EPS*MAX( ABS( TL( 1, 1 ) ), ABS( TR( 1, 1 ) ),
$ ABS( TR( 1, 2 ) ), ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ),
$ SMLNUM )
TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
TMP( 4 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
IF( LTRANR ) THEN
TMP( 2 ) = SGN*TR( 2, 1 )
TMP( 3 ) = SGN*TR( 1, 2 )
ELSE
TMP( 2 ) = SGN*TR( 1, 2 )
TMP( 3 ) = SGN*TR( 2, 1 )
END IF
BTMP( 1 ) = B( 1, 1 )
BTMP( 2 ) = B( 1, 2 )
GO TO 40
*
* 2 by 1:
* op[TL11 TL12]*[X11] + ISGN* [X11]*TR11 = [B11]
* [TL21 TL22] [X21] [X21] [B21]
*
30 CONTINUE
SMIN = MAX( EPS*MAX( ABS( TR( 1, 1 ) ), ABS( TL( 1, 1 ) ),
$ ABS( TL( 1, 2 ) ), ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ),
$ SMLNUM )
TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
TMP( 4 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
IF( LTRANL ) THEN
TMP( 2 ) = TL( 1, 2 )
TMP( 3 ) = TL( 2, 1 )
ELSE
TMP( 2 ) = TL( 2, 1 )
TMP( 3 ) = TL( 1, 2 )
END IF
BTMP( 1 ) = B( 1, 1 )
BTMP( 2 ) = B( 2, 1 )
40 CONTINUE
*
* Solve 2 by 2 system using complete pivoting.
* Set pivots less than SMIN to SMIN.
*
IPIV = IDAMAX( 4, TMP, 1 )
U11 = TMP( IPIV )
IF( ABS( U11 ).LE.SMIN ) THEN
INFO = 1
U11 = SMIN
END IF
U12 = TMP( LOCU12( IPIV ) )
L21 = TMP( LOCL21( IPIV ) ) / U11
U22 = TMP( LOCU22( IPIV ) ) - U12*L21
XSWAP = XSWPIV( IPIV )
BSWAP = BSWPIV( IPIV )
IF( ABS( U22 ).LE.SMIN ) THEN
INFO = 1
U22 = SMIN
END IF
IF( BSWAP ) THEN
TEMP = BTMP( 2 )
BTMP( 2 ) = BTMP( 1 ) - L21*TEMP
BTMP( 1 ) = TEMP
ELSE
BTMP( 2 ) = BTMP( 2 ) - L21*BTMP( 1 )
END IF
SCALE = ONE
IF( ( TWO*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( U22 ) .OR.
$ ( TWO*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( U11 ) ) THEN
SCALE = HALF / MAX( ABS( BTMP( 1 ) ), ABS( BTMP( 2 ) ) )
BTMP( 1 ) = BTMP( 1 )*SCALE
BTMP( 2 ) = BTMP( 2 )*SCALE
END IF
X2( 2 ) = BTMP( 2 ) / U22
X2( 1 ) = BTMP( 1 ) / U11 - ( U12 / U11 )*X2( 2 )
IF( XSWAP ) THEN
TEMP = X2( 2 )
X2( 2 ) = X2( 1 )
X2( 1 ) = TEMP
END IF
X( 1, 1 ) = X2( 1 )
IF( N1.EQ.1 ) THEN
X( 1, 2 ) = X2( 2 )
XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
ELSE
X( 2, 1 ) = X2( 2 )
XNORM = MAX( ABS( X( 1, 1 ) ), ABS( X( 2, 1 ) ) )
END IF
RETURN
*
* 2 by 2:
* op[TL11 TL12]*[X11 X12] +ISGN* [X11 X12]*op[TR11 TR12] = [B11 B12]
* [TL21 TL22] [X21 X22] [X21 X22] [TR21 TR22] [B21 B22]
*
* Solve equivalent 4 by 4 system using complete pivoting.
* Set pivots less than SMIN to SMIN.
*
50 CONTINUE
SMIN = MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ),
$ ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) )
SMIN = MAX( SMIN, ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ),
$ ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) )
SMIN = MAX( EPS*SMIN, SMLNUM )
BTMP( 1 ) = ZERO
CALL DCOPY( 16, BTMP, 0, T16, 1 )
T16( 1, 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
T16( 2, 2 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
T16( 3, 3 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
T16( 4, 4 ) = TL( 2, 2 ) + SGN*TR( 2, 2 )
IF( LTRANL ) THEN
T16( 1, 2 ) = TL( 2, 1 )
T16( 2, 1 ) = TL( 1, 2 )
T16( 3, 4 ) = TL( 2, 1 )
T16( 4, 3 ) = TL( 1, 2 )
ELSE
T16( 1, 2 ) = TL( 1, 2 )
T16( 2, 1 ) = TL( 2, 1 )
T16( 3, 4 ) = TL( 1, 2 )
T16( 4, 3 ) = TL( 2, 1 )
END IF
IF( LTRANR ) THEN
T16( 1, 3 ) = SGN*TR( 1, 2 )
T16( 2, 4 ) = SGN*TR( 1, 2 )
T16( 3, 1 ) = SGN*TR( 2, 1 )
T16( 4, 2 ) = SGN*TR( 2, 1 )
ELSE
T16( 1, 3 ) = SGN*TR( 2, 1 )
T16( 2, 4 ) = SGN*TR( 2, 1 )
T16( 3, 1 ) = SGN*TR( 1, 2 )
T16( 4, 2 ) = SGN*TR( 1, 2 )
END IF
BTMP( 1 ) = B( 1, 1 )
BTMP( 2 ) = B( 2, 1 )
BTMP( 3 ) = B( 1, 2 )
BTMP( 4 ) = B( 2, 2 )
*
* Perform elimination
*
DO 100 I = 1, 3
XMAX = ZERO
DO 70 IP = I, 4
DO 60 JP = I, 4
IF( ABS( T16( IP, JP ) ).GE.XMAX ) THEN
XMAX = ABS( T16( IP, JP ) )
IPSV = IP
JPSV = JP
END IF
60 CONTINUE
70 CONTINUE
IF( IPSV.NE.I ) THEN
CALL DSWAP( 4, T16( IPSV, 1 ), 4, T16( I, 1 ), 4 )
TEMP = BTMP( I )
BTMP( I ) = BTMP( IPSV )
BTMP( IPSV ) = TEMP
END IF
IF( JPSV.NE.I )
$ CALL DSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 )
JPIV( I ) = JPSV
IF( ABS( T16( I, I ) ).LT.SMIN ) THEN
INFO = 1
T16( I, I ) = SMIN
END IF
DO 90 J = I + 1, 4
T16( J, I ) = T16( J, I ) / T16( I, I )
BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I )
DO 80 K = I + 1, 4
T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K )
80 CONTINUE
90 CONTINUE
100 CONTINUE
IF( ABS( T16( 4, 4 ) ).LT.SMIN ) THEN
INFO = 1
T16( 4, 4 ) = SMIN
END IF
SCALE = ONE
IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR.
$ ( EIGHT*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T16( 2, 2 ) ) .OR.
$ ( EIGHT*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T16( 3, 3 ) ) .OR.
$ ( EIGHT*SMLNUM )*ABS( BTMP( 4 ) ).GT.ABS( T16( 4, 4 ) ) ) THEN
SCALE = ( ONE / EIGHT ) / MAX( ABS( BTMP( 1 ) ),
$ ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ), ABS( BTMP( 4 ) ) )
BTMP( 1 ) = BTMP( 1 )*SCALE
BTMP( 2 ) = BTMP( 2 )*SCALE
BTMP( 3 ) = BTMP( 3 )*SCALE
BTMP( 4 ) = BTMP( 4 )*SCALE
END IF
DO 120 I = 1, 4
K = 5 - I
TEMP = ONE / T16( K, K )
TMP( K ) = BTMP( K )*TEMP
DO 110 J = K + 1, 4
TMP( K ) = TMP( K ) - ( TEMP*T16( K, J ) )*TMP( J )
110 CONTINUE
120 CONTINUE
DO 130 I = 1, 3
IF( JPIV( 4-I ).NE.4-I ) THEN
TEMP = TMP( 4-I )
TMP( 4-I ) = TMP( JPIV( 4-I ) )
TMP( JPIV( 4-I ) ) = TEMP
END IF
130 CONTINUE
X( 1, 1 ) = TMP( 1 )
X( 2, 1 ) = TMP( 2 )
X( 1, 2 ) = TMP( 3 )
X( 2, 2 ) = TMP( 4 )
XNORM = MAX( ABS( TMP( 1 ) )+ABS( TMP( 3 ) ),
$ ABS( TMP( 2 ) )+ABS( TMP( 4 ) ) )
RETURN
*
* End of DLASY2
*
END
*> \brief \b DORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined by sgeqrf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORG2R + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORG2R( M, N, K, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORG2R generates an m by n real matrix Q with orthonormal columns,
*> which is defined as the first n columns of a product of k elementary
*> reflectors of order m
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines the
*> matrix Q. N >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the i-th column must contain the vector which
*> defines the elementary reflector H(i), for i = 1,2,...,k, as
*> returned by DGEQRF in the first k columns of its array
*> argument A.
*> On exit, the m-by-n matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORG2R( M, N, K, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, L
* ..
* .. External Subroutines ..
EXTERNAL DLARF, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORG2R', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
* Initialise columns k+1:n to columns of the unit matrix
*
DO 20 J = K + 1, N
DO 10 L = 1, M
A( L, J ) = ZERO
10 CONTINUE
A( J, J ) = ONE
20 CONTINUE
*
DO 40 I = K, 1, -1
*
* Apply H(i) to A(i:m,i:n) from the left
*
IF( I.LT.N ) THEN
A( I, I ) = ONE
CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAU( I ),
$ A( I, I+1 ), LDA, WORK )
END IF
IF( I.LT.M )
$ CALL DSCAL( M-I, -TAU( I ), A( I+1, I ), 1 )
A( I, I ) = ONE - TAU( I )
*
* Set A(1:i-1,i) to zero
*
DO 30 L = 1, I - 1
A( L, I ) = ZERO
30 CONTINUE
40 CONTINUE
RETURN
*
* End of DORG2R
*
END
*> \brief \b DORGHR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGHR + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGHR( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, ILO, INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGHR generates a real orthogonal matrix Q which is defined as the
*> product of IHI-ILO elementary reflectors of order N, as returned by
*> DGEHRD:
*>
*> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix Q. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*>
*> ILO and IHI must have the same values as in the previous call
*> of DGEHRD. Q is equal to the unit matrix except in the
*> submatrix Q(ilo+1:ihi,ilo+1:ihi).
*> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the vectors which define the elementary reflectors,
*> as returned by DGEHRD.
*> On exit, the N-by-N orthogonal matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEHRD.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= IHI-ILO.
*> For optimum performance LWORK >= (IHI-ILO)*NB, where NB is
*> the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORGHR( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IINFO, J, LWKOPT, NB, NH
* ..
* .. External Subroutines ..
EXTERNAL DORGQR, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
NH = IHI - ILO
LQUERY = ( LWORK.EQ.-1 )
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -2
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LWORK.LT.MAX( 1, NH ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
*
IF( INFO.EQ.0 ) THEN
NB = ILAENV( 1, 'DORGQR', ' ', NH, NH, NH, -1 )
LWKOPT = MAX( 1, NH )*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGHR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
* Shift the vectors which define the elementary reflectors one
* column to the right, and set the first ilo and the last n-ihi
* rows and columns to those of the unit matrix
*
DO 40 J = IHI, ILO + 1, -1
DO 10 I = 1, J - 1
A( I, J ) = ZERO
10 CONTINUE
DO 20 I = J + 1, IHI
A( I, J ) = A( I, J-1 )
20 CONTINUE
DO 30 I = IHI + 1, N
A( I, J ) = ZERO
30 CONTINUE
40 CONTINUE
DO 60 J = 1, ILO
DO 50 I = 1, N
A( I, J ) = ZERO
50 CONTINUE
A( J, J ) = ONE
60 CONTINUE
DO 80 J = IHI + 1, N
DO 70 I = 1, N
A( I, J ) = ZERO
70 CONTINUE
A( J, J ) = ONE
80 CONTINUE
*
IF( NH.GT.0 ) THEN
*
* Generate Q(ilo+1:ihi,ilo+1:ihi)
*
CALL DORGQR( NH, NH, NH, A( ILO+1, ILO+1 ), LDA, TAU( ILO ),
$ WORK, LWORK, IINFO )
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORGHR
*
END
*> \brief \b DORGQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGQR + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGQR generates an M-by-N real matrix Q with orthonormal columns,
*> which is defined as the first N columns of a product of K elementary
*> reflectors of order M
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines the
*> matrix Q. N >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the i-th column must contain the vector which
*> defines the elementary reflector H(i), for i = 1,2,...,k, as
*> returned by DGEQRF in the first k columns of its array
*> argument A.
*> On exit, the M-by-N matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N).
*> For optimum performance LWORK >= N*NB, where NB is the
*> optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, J, KI, KK, L, LDWORK,
$ LWKOPT, NB, NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORG2R, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
NB = ILAENV( 1, 'DORGQR', ' ', M, N, K, -1 )
LWKOPT = MAX( 1, N )*NB
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGQR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
NX = 0
IWS = N
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DORGQR', ' ', M, N, K, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORGQR', ' ', M, N, K, -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code after the last block.
* The first kk columns are handled by the block method.
*
KI = ( ( K-NX-1 ) / NB )*NB
KK = MIN( K, KI+NB )
*
* Set A(1:kk,kk+1:n) to zero.
*
DO 20 J = KK + 1, N
DO 10 I = 1, KK
A( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
KK = 0
END IF
*
* Use unblocked code for the last or only block.
*
IF( KK.LT.N )
$ CALL DORG2R( M-KK, N-KK, K-KK, A( KK+1, KK+1 ), LDA,
$ TAU( KK+1 ), WORK, IINFO )
*
IF( KK.GT.0 ) THEN
*
* Use blocked code
*
DO 50 I = KI + 1, 1, -NB
IB = MIN( NB, K-I+1 )
IF( I+IB.LE.N ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Columnwise', M-I+1, IB,
$ A( I, I ), LDA, TAU( I ), WORK, LDWORK )
*
* Apply H to A(i:m,i+ib:n) from the left
*
CALL DLARFB( 'Left', 'No transpose', 'Forward',
$ 'Columnwise', M-I+1, N-I-IB+1, IB,
$ A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ),
$ LDA, WORK( IB+1 ), LDWORK )
END IF
*
* Apply H to rows i:m of current block
*
CALL DORG2R( M-I+1, IB, IB, A( I, I ), LDA, TAU( I ), WORK,
$ IINFO )
*
* Set rows 1:i-1 of current block to zero
*
DO 40 J = I, I + IB - 1
DO 30 L = 1, I - 1
A( L, J ) = ZERO
30 CONTINUE
40 CONTINUE
50 CONTINUE
END IF
*
WORK( 1 ) = IWS
RETURN
*
* End of DORGQR
*
END
*> \brief \b DORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sgeqrf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORM2R + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORM2R overwrites the general real m by n matrix C with
*>
*> Q * C if SIDE = 'L' and TRANS = 'N', or
*>
*> Q**T* C if SIDE = 'L' and TRANS = 'T', or
*>
*> C * Q if SIDE = 'R' and TRANS = 'N', or
*>
*> C * Q**T if SIDE = 'R' and TRANS = 'T',
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF. Q is of order m if SIDE = 'L' and of order n
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left
*> = 'R': apply Q or Q**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': apply Q (No transpose)
*> = 'T': apply Q**T (Transpose)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,K)
*> The i-th column must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGEQRF in the first k columns of its array argument A.
*> A is modified by the routine but restored on exit.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If SIDE = 'L', LDA >= max(1,M);
*> if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L',
*> (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LEFT, NOTRAN
INTEGER I, I1, I2, I3, IC, JC, MI, NI, NQ
DOUBLE PRECISION AII
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLARF, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
*
* NQ is the order of Q
*
IF( LEFT ) THEN
NQ = M
ELSE
NQ = N
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORM2R', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 )
$ RETURN
*
IF( ( LEFT .AND. .NOT.NOTRAN ) .OR. ( .NOT.LEFT .AND. NOTRAN ) )
$ THEN
I1 = 1
I2 = K
I3 = 1
ELSE
I1 = K
I2 = 1
I3 = -1
END IF
*
IF( LEFT ) THEN
NI = N
JC = 1
ELSE
MI = M
IC = 1
END IF
*
DO 10 I = I1, I2, I3
IF( LEFT ) THEN
*
* H(i) is applied to C(i:m,1:n)
*
MI = M - I + 1
IC = I
ELSE
*
* H(i) is applied to C(1:m,i:n)
*
NI = N - I + 1
JC = I
END IF
*
* Apply H(i)
*
AII = A( I, I )
A( I, I ) = ONE
CALL DLARF( SIDE, MI, NI, A( I, I ), 1, TAU( I ), C( IC, JC ),
$ LDC, WORK )
A( I, I ) = AII
10 CONTINUE
RETURN
*
* End of DORM2R
*
END
*> \brief \b DORMHR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMHR + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMHR( SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C,
* LDC, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER IHI, ILO, INFO, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMHR overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> where Q is a real orthogonal matrix of order nq, with nq = m if
*> SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
*> IHI-ILO elementary reflectors, as returned by DGEHRD:
*>
*> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*>
*> ILO and IHI must have the same values as in the previous call
*> of DGEHRD. Q is equal to the unit matrix except in the
*> submatrix Q(ilo+1:ihi,ilo+1:ihi).
*> If SIDE = 'L', then 1 <= ILO <= IHI <= M, if M > 0, and
*> ILO = 1 and IHI = 0, if M = 0;
*> if SIDE = 'R', then 1 <= ILO <= IHI <= N, if N > 0, and
*> ILO = 1 and IHI = 0, if N = 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,M) if SIDE = 'L'
*> (LDA,N) if SIDE = 'R'
*> The vectors which define the elementary reflectors, as
*> returned by DGEHRD.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension
*> (M-1) if SIDE = 'L'
*> (N-1) if SIDE = 'R'
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEHRD.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For optimum performance LWORK >= N*NB if SIDE = 'L', and
*> LWORK >= M*NB if SIDE = 'R', where NB is the optimal
*> blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORMHR( SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C,
$ LDC, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER IHI, ILO, INFO, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LEFT, LQUERY
INTEGER I1, I2, IINFO, LWKOPT, MI, NB, NH, NI, NQ, NW
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DORMQR, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
NH = IHI - ILO
LEFT = LSAME( SIDE, 'L' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = N
ELSE
NQ = N
NW = M
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT.LSAME( TRANS, 'T' ) )
$ THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, NQ ) ) THEN
INFO = -5
ELSE IF( IHI.LT.MIN( ILO, NQ ) .OR. IHI.GT.NQ ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -11
ELSE IF( LWORK.LT.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
*
IF( INFO.EQ.0 ) THEN
IF( LEFT ) THEN
NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, NH, N, NH, -1 )
ELSE
NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M, NH, NH, -1 )
END IF
LWKOPT = MAX( 1, NW )*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMHR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. NH.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
IF( LEFT ) THEN
MI = NH
NI = N
I1 = ILO + 1
I2 = 1
ELSE
MI = M
NI = NH
I1 = 1
I2 = ILO + 1
END IF
*
CALL DORMQR( SIDE, TRANS, MI, NI, NH, A( ILO+1, ILO ), LDA,
$ TAU( ILO ), C( I1, I2 ), LDC, WORK, LWORK, IINFO )
*
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORMHR
*
END
*> \brief \b DORMQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMQR + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMQR( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMQR overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF. Q is of order M if SIDE = 'L' and of order N
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,K)
*> The i-th column must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGEQRF in the first k columns of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If SIDE = 'L', LDA >= max(1,M);
*> if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For good performance, LWORK should generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2015
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORMQR( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2015
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER NBMAX, LDT, TSIZE
PARAMETER ( NBMAX = 64, LDT = NBMAX+1,
$ TSIZE = LDT*NBMAX )
* ..
* .. Local Scalars ..
LOGICAL LEFT, LQUERY, NOTRAN
INTEGER I, I1, I2, I3, IB, IC, IINFO, IWT, JC, LDWORK,
$ LWKOPT, MI, NB, NBMIN, NI, NQ, NW
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORM2R, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = N
ELSE
NQ = N
NW = M
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LWORK.LT.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Compute the workspace requirements
*
NB = MIN( NBMAX, ILAENV( 1, 'DORMQR', SIDE // TRANS, M, N, K,
$ -1 ) )
LWKOPT = MAX( 1, NW )*NB + TSIZE
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMQR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
LDWORK = NW
IF( NB.GT.1 .AND. NB.LT.K ) THEN
IF( LWORK.LT.NW*NB+TSIZE ) THEN
NB = (LWORK-TSIZE) / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORMQR', SIDE // TRANS, M, N, K,
$ -1 ) )
END IF
END IF
*
IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
* Use unblocked code
*
CALL DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK,
$ IINFO )
ELSE
*
* Use blocked code
*
IWT = 1 + NW*NB
IF( ( LEFT .AND. .NOT.NOTRAN ) .OR.
$ ( .NOT.LEFT .AND. NOTRAN ) ) THEN
I1 = 1
I2 = K
I3 = NB
ELSE
I1 = ( ( K-1 ) / NB )*NB + 1
I2 = 1
I3 = -NB
END IF
*
IF( LEFT ) THEN
NI = N
JC = 1
ELSE
MI = M
IC = 1
END IF
*
DO 10 I = I1, I2, I3
IB = MIN( NB, K-I+1 )
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Columnwise', NQ-I+1, IB, A( I, I ),
$ LDA, TAU( I ), WORK( IWT ), LDT )
IF( LEFT ) THEN
*
* H or H**T is applied to C(i:m,1:n)
*
MI = M - I + 1
IC = I
ELSE
*
* H or H**T is applied to C(1:m,i:n)
*
NI = N - I + 1
JC = I
END IF
*
* Apply H or H**T
*
CALL DLARFB( SIDE, TRANS, 'Forward', 'Columnwise', MI, NI,
$ IB, A( I, I ), LDA, WORK( IWT ), LDT,
$ C( IC, JC ), LDC, WORK, LDWORK )
10 CONTINUE
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORMQR
*
END
*> \brief \b DTREVC
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTREVC + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
* LDVR, MM, M, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER HOWMNY, SIDE
* INTEGER INFO, LDT, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
* LOGICAL SELECT( * )
* DOUBLE PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTREVC computes some or all of the right and/or left eigenvectors of
*> a real upper quasi-triangular matrix T.
*> Matrices of this type are produced by the Schur factorization of
*> a real general matrix: A = Q*T*Q**T, as computed by DHSEQR.
*>
*> The right eigenvector x and the left eigenvector y of T corresponding
*> to an eigenvalue w are defined by:
*>
*> T*x = w*x, (y**T)*T = w*(y**T)
*>
*> where y**T denotes the transpose of y.
*> The eigenvalues are not input to this routine, but are read directly
*> from the diagonal blocks of T.
*>
*> This routine returns the matrices X and/or Y of right and left
*> eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
*> input matrix. If Q is the orthogonal factor that reduces a matrix
*> A to Schur form T, then Q*X and Q*Y are the matrices of right and
*> left eigenvectors of A.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'R': compute right eigenvectors only;
*> = 'L': compute left eigenvectors only;
*> = 'B': compute both right and left eigenvectors.
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*> HOWMNY is CHARACTER*1
*> = 'A': compute all right and/or left eigenvectors;
*> = 'B': compute all right and/or left eigenvectors,
*> backtransformed by the matrices in VR and/or VL;
*> = 'S': compute selected right and/or left eigenvectors,
*> as indicated by the logical array SELECT.
*> \endverbatim
*>
*> \param[in,out] SELECT
*> \verbatim
*> SELECT is LOGICAL array, dimension (N)
*> If HOWMNY = 'S', SELECT specifies the eigenvectors to be
*> computed.
*> If w(j) is a real eigenvalue, the corresponding real
*> eigenvector is computed if SELECT(j) is .TRUE..
*> If w(j) and w(j+1) are the real and imaginary parts of a
*> complex eigenvalue, the corresponding complex eigenvector is
*> computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
*> on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
*> .FALSE..
*> Not referenced if HOWMNY = 'A' or 'B'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> The upper quasi-triangular matrix T in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,MM)
*> On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
*> contain an N-by-N matrix Q (usually the orthogonal matrix Q
*> of Schur vectors returned by DHSEQR).
*> On exit, if SIDE = 'L' or 'B', VL contains:
*> if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
*> if HOWMNY = 'B', the matrix Q*Y;
*> if HOWMNY = 'S', the left eigenvectors of T specified by
*> SELECT, stored consecutively in the columns
*> of VL, in the same order as their
*> eigenvalues.
*> A complex eigenvector corresponding to a complex eigenvalue
*> is stored in two consecutive columns, the first holding the
*> real part, and the second the imaginary part.
*> Not referenced if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the array VL. LDVL >= 1, and if
*> SIDE = 'L' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,MM)
*> On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
*> contain an N-by-N matrix Q (usually the orthogonal matrix Q
*> of Schur vectors returned by DHSEQR).
*> On exit, if SIDE = 'R' or 'B', VR contains:
*> if HOWMNY = 'A', the matrix X of right eigenvectors of T;
*> if HOWMNY = 'B', the matrix Q*X;
*> if HOWMNY = 'S', the right eigenvectors of T specified by
*> SELECT, stored consecutively in the columns
*> of VR, in the same order as their
*> eigenvalues.
*> A complex eigenvector corresponding to a complex eigenvalue
*> is stored in two consecutive columns, the first holding the
*> real part and the second the imaginary part.
*> Not referenced if SIDE = 'L'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the array VR. LDVR >= 1, and if
*> SIDE = 'R' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*> MM is INTEGER
*> The number of columns in the arrays VL and/or VR. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The number of columns in the arrays VL and/or VR actually
*> used to store the eigenvectors.
*> If HOWMNY = 'A' or 'B', M is set to N.
*> Each selected real eigenvector occupies one column and each
*> selected complex eigenvector occupies two columns.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The algorithm used in this program is basically backward (forward)
*> substitution, with scaling to make the the code robust against
*> possible overflow.
*>
*> Each eigenvector is normalized so that the element of largest
*> magnitude has magnitude 1; here the magnitude of a complex number
*> (x,y) is taken to be |x| + |y|.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
$ LDVR, MM, M, WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER HOWMNY, SIDE
INTEGER INFO, LDT, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
DOUBLE PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ALLV, BOTHV, LEFTV, OVER, PAIR, RIGHTV, SOMEV
INTEGER I, IERR, II, IP, IS, J, J1, J2, JNXT, K, KI, N2
DOUBLE PRECISION BETA, BIGNUM, EMAX, OVFL, REC, REMAX, SCALE,
$ SMIN, SMLNUM, ULP, UNFL, VCRIT, VMAX, WI, WR,
$ XNORM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DDOT, DLAMCH
EXTERNAL LSAME, IDAMAX, DDOT, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMV, DLALN2, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Local Arrays ..
DOUBLE PRECISION X( 2, 2 )
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
BOTHV = LSAME( SIDE, 'B' )
RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV
*
ALLV = LSAME( HOWMNY, 'A' )
OVER = LSAME( HOWMNY, 'B' )
SOMEV = LSAME( HOWMNY, 'S' )
*
INFO = 0
IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
INFO = -1
ELSE IF( .NOT.ALLV .AND. .NOT.OVER .AND. .NOT.SOMEV ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN
INFO = -8
ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN
INFO = -10
ELSE
*
* Set M to the number of columns required to store the selected
* eigenvectors, standardize the array SELECT if necessary, and
* test MM.
*
IF( SOMEV ) THEN
M = 0
PAIR = .FALSE.
DO 10 J = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
SELECT( J ) = .FALSE.
ELSE
IF( J.LT.N ) THEN
IF( T( J+1, J ).EQ.ZERO ) THEN
IF( SELECT( J ) )
$ M = M + 1
ELSE
PAIR = .TRUE.
IF( SELECT( J ) .OR. SELECT( J+1 ) ) THEN
SELECT( J ) = .TRUE.
M = M + 2
END IF
END IF
ELSE
IF( SELECT( N ) )
$ M = M + 1
END IF
END IF
10 CONTINUE
ELSE
M = N
END IF
*
IF( MM.LT.M ) THEN
INFO = -11
END IF
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTREVC', -INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( N.EQ.0 )
$ RETURN
*
* Set the constants to control overflow.
*
UNFL = DLAMCH( 'Safe minimum' )
OVFL = ONE / UNFL
CALL DLABAD( UNFL, OVFL )
ULP = DLAMCH( 'Precision' )
SMLNUM = UNFL*( N / ULP )
BIGNUM = ( ONE-ULP ) / SMLNUM
*
* Compute 1-norm of each column of strictly upper triangular
* part of T to control overflow in triangular solver.
*
WORK( 1 ) = ZERO
DO 30 J = 2, N
WORK( J ) = ZERO
DO 20 I = 1, J - 1
WORK( J ) = WORK( J ) + ABS( T( I, J ) )
20 CONTINUE
30 CONTINUE
*
* Index IP is used to specify the real or complex eigenvalue:
* IP = 0, real eigenvalue,
* 1, first of conjugate complex pair: (wr,wi)
* -1, second of conjugate complex pair: (wr,wi)
*
N2 = 2*N
*
IF( RIGHTV ) THEN
*
* Compute right eigenvectors.
*
IP = 0
IS = M
DO 140 KI = N, 1, -1
*
IF( IP.EQ.1 )
$ GO TO 130
IF( KI.EQ.1 )
$ GO TO 40
IF( T( KI, KI-1 ).EQ.ZERO )
$ GO TO 40
IP = -1
*
40 CONTINUE
IF( SOMEV ) THEN
IF( IP.EQ.0 ) THEN
IF( .NOT.SELECT( KI ) )
$ GO TO 130
ELSE
IF( .NOT.SELECT( KI-1 ) )
$ GO TO 130
END IF
END IF
*
* Compute the KI-th eigenvalue (WR,WI).
*
WR = T( KI, KI )
WI = ZERO
IF( IP.NE.0 )
$ WI = SQRT( ABS( T( KI, KI-1 ) ) )*
$ SQRT( ABS( T( KI-1, KI ) ) )
SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM )
*
IF( IP.EQ.0 ) THEN
*
* Real right eigenvector
*
WORK( KI+N ) = ONE
*
* Form right-hand side
*
DO 50 K = 1, KI - 1
WORK( K+N ) = -T( K, KI )
50 CONTINUE
*
* Solve the upper quasi-triangular system:
* (T(1:KI-1,1:KI-1) - WR)*X = SCALE*WORK.
*
JNXT = KI - 1
DO 60 J = KI - 1, 1, -1
IF( J.GT.JNXT )
$ GO TO 60
J1 = J
J2 = J
JNXT = J - 1
IF( J.GT.1 ) THEN
IF( T( J, J-1 ).NE.ZERO ) THEN
J1 = J - 1
JNXT = J - 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1-by-1 diagonal block
*
CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J, J ),
$ LDT, ONE, ONE, WORK( J+N ), N, WR,
$ ZERO, X, 2, SCALE, XNORM, IERR )
*
* Scale X(1,1) to avoid overflow when updating
* the right-hand side.
*
IF( XNORM.GT.ONE ) THEN
IF( WORK( J ).GT.BIGNUM / XNORM ) THEN
X( 1, 1 ) = X( 1, 1 ) / XNORM
SCALE = SCALE / XNORM
END IF
END IF
*
* Scale if necessary
*
IF( SCALE.NE.ONE )
$ CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 )
WORK( J+N ) = X( 1, 1 )
*
* Update right-hand side
*
CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1,
$ WORK( 1+N ), 1 )
*
ELSE
*
* 2-by-2 diagonal block
*
CALL DLALN2( .FALSE., 2, 1, SMIN, ONE,
$ T( J-1, J-1 ), LDT, ONE, ONE,
$ WORK( J-1+N ), N, WR, ZERO, X, 2,
$ SCALE, XNORM, IERR )
*
* Scale X(1,1) and X(2,1) to avoid overflow when
* updating the right-hand side.
*
IF( XNORM.GT.ONE ) THEN
BETA = MAX( WORK( J-1 ), WORK( J ) )
IF( BETA.GT.BIGNUM / XNORM ) THEN
X( 1, 1 ) = X( 1, 1 ) / XNORM
X( 2, 1 ) = X( 2, 1 ) / XNORM
SCALE = SCALE / XNORM
END IF
END IF
*
* Scale if necessary
*
IF( SCALE.NE.ONE )
$ CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 )
WORK( J-1+N ) = X( 1, 1 )
WORK( J+N ) = X( 2, 1 )
*
* Update right-hand side
*
CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1,
$ WORK( 1+N ), 1 )
CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1,
$ WORK( 1+N ), 1 )
END IF
60 CONTINUE
*
* Copy the vector x or Q*x to VR and normalize.
*
IF( .NOT.OVER ) THEN
CALL DCOPY( KI, WORK( 1+N ), 1, VR( 1, IS ), 1 )
*
II = IDAMAX( KI, VR( 1, IS ), 1 )
REMAX = ONE / ABS( VR( II, IS ) )
CALL DSCAL( KI, REMAX, VR( 1, IS ), 1 )
*
DO 70 K = KI + 1, N
VR( K, IS ) = ZERO
70 CONTINUE
ELSE
IF( KI.GT.1 )
$ CALL DGEMV( 'N', N, KI-1, ONE, VR, LDVR,
$ WORK( 1+N ), 1, WORK( KI+N ),
$ VR( 1, KI ), 1 )
*
II = IDAMAX( N, VR( 1, KI ), 1 )
REMAX = ONE / ABS( VR( II, KI ) )
CALL DSCAL( N, REMAX, VR( 1, KI ), 1 )
END IF
*
ELSE
*
* Complex right eigenvector.
*
* Initial solve
* [ (T(KI-1,KI-1) T(KI-1,KI) ) - (WR + I* WI)]*X = 0.
* [ (T(KI,KI-1) T(KI,KI) ) ]
*
IF( ABS( T( KI-1, KI ) ).GE.ABS( T( KI, KI-1 ) ) ) THEN
WORK( KI-1+N ) = ONE
WORK( KI+N2 ) = WI / T( KI-1, KI )
ELSE
WORK( KI-1+N ) = -WI / T( KI, KI-1 )
WORK( KI+N2 ) = ONE
END IF
WORK( KI+N ) = ZERO
WORK( KI-1+N2 ) = ZERO
*
* Form right-hand side
*
DO 80 K = 1, KI - 2
WORK( K+N ) = -WORK( KI-1+N )*T( K, KI-1 )
WORK( K+N2 ) = -WORK( KI+N2 )*T( K, KI )
80 CONTINUE
*
* Solve upper quasi-triangular system:
* (T(1:KI-2,1:KI-2) - (WR+i*WI))*X = SCALE*(WORK+i*WORK2)
*
JNXT = KI - 2
DO 90 J = KI - 2, 1, -1
IF( J.GT.JNXT )
$ GO TO 90
J1 = J
J2 = J
JNXT = J - 1
IF( J.GT.1 ) THEN
IF( T( J, J-1 ).NE.ZERO ) THEN
J1 = J - 1
JNXT = J - 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1-by-1 diagonal block
*
CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J, J ),
$ LDT, ONE, ONE, WORK( J+N ), N, WR, WI,
$ X, 2, SCALE, XNORM, IERR )
*
* Scale X(1,1) and X(1,2) to avoid overflow when
* updating the right-hand side.
*
IF( XNORM.GT.ONE ) THEN
IF( WORK( J ).GT.BIGNUM / XNORM ) THEN
X( 1, 1 ) = X( 1, 1 ) / XNORM
X( 1, 2 ) = X( 1, 2 ) / XNORM
SCALE = SCALE / XNORM
END IF
END IF
*
* Scale if necessary
*
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 )
CALL DSCAL( KI, SCALE, WORK( 1+N2 ), 1 )
END IF
WORK( J+N ) = X( 1, 1 )
WORK( J+N2 ) = X( 1, 2 )
*
* Update the right-hand side
*
CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1,
$ WORK( 1+N ), 1 )
CALL DAXPY( J-1, -X( 1, 2 ), T( 1, J ), 1,
$ WORK( 1+N2 ), 1 )
*
ELSE
*
* 2-by-2 diagonal block
*
CALL DLALN2( .FALSE., 2, 2, SMIN, ONE,
$ T( J-1, J-1 ), LDT, ONE, ONE,
$ WORK( J-1+N ), N, WR, WI, X, 2, SCALE,
$ XNORM, IERR )
*
* Scale X to avoid overflow when updating
* the right-hand side.
*
IF( XNORM.GT.ONE ) THEN
BETA = MAX( WORK( J-1 ), WORK( J ) )
IF( BETA.GT.BIGNUM / XNORM ) THEN
REC = ONE / XNORM
X( 1, 1 ) = X( 1, 1 )*REC
X( 1, 2 ) = X( 1, 2 )*REC
X( 2, 1 ) = X( 2, 1 )*REC
X( 2, 2 ) = X( 2, 2 )*REC
SCALE = SCALE*REC
END IF
END IF
*
* Scale if necessary
*
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 )
CALL DSCAL( KI, SCALE, WORK( 1+N2 ), 1 )
END IF
WORK( J-1+N ) = X( 1, 1 )
WORK( J+N ) = X( 2, 1 )
WORK( J-1+N2 ) = X( 1, 2 )
WORK( J+N2 ) = X( 2, 2 )
*
* Update the right-hand side
*
CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1,
$ WORK( 1+N ), 1 )
CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1,
$ WORK( 1+N ), 1 )
CALL DAXPY( J-2, -X( 1, 2 ), T( 1, J-1 ), 1,
$ WORK( 1+N2 ), 1 )
CALL DAXPY( J-2, -X( 2, 2 ), T( 1, J ), 1,
$ WORK( 1+N2 ), 1 )
END IF
90 CONTINUE
*
* Copy the vector x or Q*x to VR and normalize.
*
IF( .NOT.OVER ) THEN
CALL DCOPY( KI, WORK( 1+N ), 1, VR( 1, IS-1 ), 1 )
CALL DCOPY( KI, WORK( 1+N2 ), 1, VR( 1, IS ), 1 )
*
EMAX = ZERO
DO 100 K = 1, KI
EMAX = MAX( EMAX, ABS( VR( K, IS-1 ) )+
$ ABS( VR( K, IS ) ) )
100 CONTINUE
*
REMAX = ONE / EMAX
CALL DSCAL( KI, REMAX, VR( 1, IS-1 ), 1 )
CALL DSCAL( KI, REMAX, VR( 1, IS ), 1 )
*
DO 110 K = KI + 1, N
VR( K, IS-1 ) = ZERO
VR( K, IS ) = ZERO
110 CONTINUE
*
ELSE
*
IF( KI.GT.2 ) THEN
CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR,
$ WORK( 1+N ), 1, WORK( KI-1+N ),
$ VR( 1, KI-1 ), 1 )
CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR,
$ WORK( 1+N2 ), 1, WORK( KI+N2 ),
$ VR( 1, KI ), 1 )
ELSE
CALL DSCAL( N, WORK( KI-1+N ), VR( 1, KI-1 ), 1 )
CALL DSCAL( N, WORK( KI+N2 ), VR( 1, KI ), 1 )
END IF
*
EMAX = ZERO
DO 120 K = 1, N
EMAX = MAX( EMAX, ABS( VR( K, KI-1 ) )+
$ ABS( VR( K, KI ) ) )
120 CONTINUE
REMAX = ONE / EMAX
CALL DSCAL( N, REMAX, VR( 1, KI-1 ), 1 )
CALL DSCAL( N, REMAX, VR( 1, KI ), 1 )
END IF
END IF
*
IS = IS - 1
IF( IP.NE.0 )
$ IS = IS - 1
130 CONTINUE
IF( IP.EQ.1 )
$ IP = 0
IF( IP.EQ.-1 )
$ IP = 1
140 CONTINUE
END IF
*
IF( LEFTV ) THEN
*
* Compute left eigenvectors.
*
IP = 0
IS = 1
DO 260 KI = 1, N
*
IF( IP.EQ.-1 )
$ GO TO 250
IF( KI.EQ.N )
$ GO TO 150
IF( T( KI+1, KI ).EQ.ZERO )
$ GO TO 150
IP = 1
*
150 CONTINUE
IF( SOMEV ) THEN
IF( .NOT.SELECT( KI ) )
$ GO TO 250
END IF
*
* Compute the KI-th eigenvalue (WR,WI).
*
WR = T( KI, KI )
WI = ZERO
IF( IP.NE.0 )
$ WI = SQRT( ABS( T( KI, KI+1 ) ) )*
$ SQRT( ABS( T( KI+1, KI ) ) )
SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM )
*
IF( IP.EQ.0 ) THEN
*
* Real left eigenvector.
*
WORK( KI+N ) = ONE
*
* Form right-hand side
*
DO 160 K = KI + 1, N
WORK( K+N ) = -T( KI, K )
160 CONTINUE
*
* Solve the quasi-triangular system:
* (T(KI+1:N,KI+1:N) - WR)**T*X = SCALE*WORK
*
VMAX = ONE
VCRIT = BIGNUM
*
JNXT = KI + 1
DO 170 J = KI + 1, N
IF( J.LT.JNXT )
$ GO TO 170
J1 = J
J2 = J
JNXT = J + 1
IF( J.LT.N ) THEN
IF( T( J+1, J ).NE.ZERO ) THEN
J2 = J + 1
JNXT = J + 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1-by-1 diagonal block
*
* Scale if necessary to avoid overflow when forming
* the right-hand side.
*
IF( WORK( J ).GT.VCRIT ) THEN
REC = ONE / VMAX
CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 )
VMAX = ONE
VCRIT = BIGNUM
END IF
*
WORK( J+N ) = WORK( J+N ) -
$ DDOT( J-KI-1, T( KI+1, J ), 1,
$ WORK( KI+1+N ), 1 )
*
* Solve (T(J,J)-WR)**T*X = WORK
*
CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J, J ),
$ LDT, ONE, ONE, WORK( J+N ), N, WR,
$ ZERO, X, 2, SCALE, XNORM, IERR )
*
* Scale if necessary
*
IF( SCALE.NE.ONE )
$ CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 )
WORK( J+N ) = X( 1, 1 )
VMAX = MAX( ABS( WORK( J+N ) ), VMAX )
VCRIT = BIGNUM / VMAX
*
ELSE
*
* 2-by-2 diagonal block
*
* Scale if necessary to avoid overflow when forming
* the right-hand side.
*
BETA = MAX( WORK( J ), WORK( J+1 ) )
IF( BETA.GT.VCRIT ) THEN
REC = ONE / VMAX
CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 )
VMAX = ONE
VCRIT = BIGNUM
END IF
*
WORK( J+N ) = WORK( J+N ) -
$ DDOT( J-KI-1, T( KI+1, J ), 1,
$ WORK( KI+1+N ), 1 )
*
WORK( J+1+N ) = WORK( J+1+N ) -
$ DDOT( J-KI-1, T( KI+1, J+1 ), 1,
$ WORK( KI+1+N ), 1 )
*
* Solve
* [T(J,J)-WR T(J,J+1) ]**T * X = SCALE*( WORK1 )
* [T(J+1,J) T(J+1,J+1)-WR] ( WORK2 )
*
CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, T( J, J ),
$ LDT, ONE, ONE, WORK( J+N ), N, WR,
$ ZERO, X, 2, SCALE, XNORM, IERR )
*
* Scale if necessary
*
IF( SCALE.NE.ONE )
$ CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 )
WORK( J+N ) = X( 1, 1 )
WORK( J+1+N ) = X( 2, 1 )
*
VMAX = MAX( ABS( WORK( J+N ) ),
$ ABS( WORK( J+1+N ) ), VMAX )
VCRIT = BIGNUM / VMAX
*
END IF
170 CONTINUE
*
* Copy the vector x or Q*x to VL and normalize.
*
IF( .NOT.OVER ) THEN
CALL DCOPY( N-KI+1, WORK( KI+N ), 1, VL( KI, IS ), 1 )
*
II = IDAMAX( N-KI+1, VL( KI, IS ), 1 ) + KI - 1
REMAX = ONE / ABS( VL( II, IS ) )
CALL DSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 )
*
DO 180 K = 1, KI - 1
VL( K, IS ) = ZERO
180 CONTINUE
*
ELSE
*
IF( KI.LT.N )
$ CALL DGEMV( 'N', N, N-KI, ONE, VL( 1, KI+1 ), LDVL,
$ WORK( KI+1+N ), 1, WORK( KI+N ),
$ VL( 1, KI ), 1 )
*
II = IDAMAX( N, VL( 1, KI ), 1 )
REMAX = ONE / ABS( VL( II, KI ) )
CALL DSCAL( N, REMAX, VL( 1, KI ), 1 )
*
END IF
*
ELSE
*
* Complex left eigenvector.
*
* Initial solve:
* ((T(KI,KI) T(KI,KI+1) )**T - (WR - I* WI))*X = 0.
* ((T(KI+1,KI) T(KI+1,KI+1)) )
*
IF( ABS( T( KI, KI+1 ) ).GE.ABS( T( KI+1, KI ) ) ) THEN
WORK( KI+N ) = WI / T( KI, KI+1 )
WORK( KI+1+N2 ) = ONE
ELSE
WORK( KI+N ) = ONE
WORK( KI+1+N2 ) = -WI / T( KI+1, KI )
END IF
WORK( KI+1+N ) = ZERO
WORK( KI+N2 ) = ZERO
*
* Form right-hand side
*
DO 190 K = KI + 2, N
WORK( K+N ) = -WORK( KI+N )*T( KI, K )
WORK( K+N2 ) = -WORK( KI+1+N2 )*T( KI+1, K )
190 CONTINUE
*
* Solve complex quasi-triangular system:
* ( T(KI+2,N:KI+2,N) - (WR-i*WI) )*X = WORK1+i*WORK2
*
VMAX = ONE
VCRIT = BIGNUM
*
JNXT = KI + 2
DO 200 J = KI + 2, N
IF( J.LT.JNXT )
$ GO TO 200
J1 = J
J2 = J
JNXT = J + 1
IF( J.LT.N ) THEN
IF( T( J+1, J ).NE.ZERO ) THEN
J2 = J + 1
JNXT = J + 2
END IF
END IF
*
IF( J1.EQ.J2 ) THEN
*
* 1-by-1 diagonal block
*
* Scale if necessary to avoid overflow when
* forming the right-hand side elements.
*
IF( WORK( J ).GT.VCRIT ) THEN
REC = ONE / VMAX
CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 )
CALL DSCAL( N-KI+1, REC, WORK( KI+N2 ), 1 )
VMAX = ONE
VCRIT = BIGNUM
END IF
*
WORK( J+N ) = WORK( J+N ) -
$ DDOT( J-KI-2, T( KI+2, J ), 1,
$ WORK( KI+2+N ), 1 )
WORK( J+N2 ) = WORK( J+N2 ) -
$ DDOT( J-KI-2, T( KI+2, J ), 1,
$ WORK( KI+2+N2 ), 1 )
*
* Solve (T(J,J)-(WR-i*WI))*(X11+i*X12)= WK+I*WK2
*
CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J, J ),
$ LDT, ONE, ONE, WORK( J+N ), N, WR,
$ -WI, X, 2, SCALE, XNORM, IERR )
*
* Scale if necessary
*
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 )
CALL DSCAL( N-KI+1, SCALE, WORK( KI+N2 ), 1 )
END IF
WORK( J+N ) = X( 1, 1 )
WORK( J+N2 ) = X( 1, 2 )
VMAX = MAX( ABS( WORK( J+N ) ),
$ ABS( WORK( J+N2 ) ), VMAX )
VCRIT = BIGNUM / VMAX
*
ELSE
*
* 2-by-2 diagonal block
*
* Scale if necessary to avoid overflow when forming
* the right-hand side elements.
*
BETA = MAX( WORK( J ), WORK( J+1 ) )
IF( BETA.GT.VCRIT ) THEN
REC = ONE / VMAX
CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 )
CALL DSCAL( N-KI+1, REC, WORK( KI+N2 ), 1 )
VMAX = ONE
VCRIT = BIGNUM
END IF
*
WORK( J+N ) = WORK( J+N ) -
$ DDOT( J-KI-2, T( KI+2, J ), 1,
$ WORK( KI+2+N ), 1 )
*
WORK( J+N2 ) = WORK( J+N2 ) -
$ DDOT( J-KI-2, T( KI+2, J ), 1,
$ WORK( KI+2+N2 ), 1 )
*
WORK( J+1+N ) = WORK( J+1+N ) -
$ DDOT( J-KI-2, T( KI+2, J+1 ), 1,
$ WORK( KI+2+N ), 1 )
*
WORK( J+1+N2 ) = WORK( J+1+N2 ) -
$ DDOT( J-KI-2, T( KI+2, J+1 ), 1,
$ WORK( KI+2+N2 ), 1 )
*
* Solve 2-by-2 complex linear equation
* ([T(j,j) T(j,j+1) ]**T-(wr-i*wi)*I)*X = SCALE*B
* ([T(j+1,j) T(j+1,j+1)] )
*
CALL DLALN2( .TRUE., 2, 2, SMIN, ONE, T( J, J ),
$ LDT, ONE, ONE, WORK( J+N ), N, WR,
$ -WI, X, 2, SCALE, XNORM, IERR )
*
* Scale if necessary
*
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 )
CALL DSCAL( N-KI+1, SCALE, WORK( KI+N2 ), 1 )
END IF
WORK( J+N ) = X( 1, 1 )
WORK( J+N2 ) = X( 1, 2 )
WORK( J+1+N ) = X( 2, 1 )
WORK( J+1+N2 ) = X( 2, 2 )
VMAX = MAX( ABS( X( 1, 1 ) ), ABS( X( 1, 2 ) ),
$ ABS( X( 2, 1 ) ), ABS( X( 2, 2 ) ), VMAX )
VCRIT = BIGNUM / VMAX
*
END IF
200 CONTINUE
*
* Copy the vector x or Q*x to VL and normalize.
*
IF( .NOT.OVER ) THEN
CALL DCOPY( N-KI+1, WORK( KI+N ), 1, VL( KI, IS ), 1 )
CALL DCOPY( N-KI+1, WORK( KI+N2 ), 1, VL( KI, IS+1 ),
$ 1 )
*
EMAX = ZERO
DO 220 K = KI, N
EMAX = MAX( EMAX, ABS( VL( K, IS ) )+
$ ABS( VL( K, IS+1 ) ) )
220 CONTINUE
REMAX = ONE / EMAX
CALL DSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 )
CALL DSCAL( N-KI+1, REMAX, VL( KI, IS+1 ), 1 )
*
DO 230 K = 1, KI - 1
VL( K, IS ) = ZERO
VL( K, IS+1 ) = ZERO
230 CONTINUE
ELSE
IF( KI.LT.N-1 ) THEN
CALL DGEMV( 'N', N, N-KI-1, ONE, VL( 1, KI+2 ),
$ LDVL, WORK( KI+2+N ), 1, WORK( KI+N ),
$ VL( 1, KI ), 1 )
CALL DGEMV( 'N', N, N-KI-1, ONE, VL( 1, KI+2 ),
$ LDVL, WORK( KI+2+N2 ), 1,
$ WORK( KI+1+N2 ), VL( 1, KI+1 ), 1 )
ELSE
CALL DSCAL( N, WORK( KI+N ), VL( 1, KI ), 1 )
CALL DSCAL( N, WORK( KI+1+N2 ), VL( 1, KI+1 ), 1 )
END IF
*
EMAX = ZERO
DO 240 K = 1, N
EMAX = MAX( EMAX, ABS( VL( K, KI ) )+
$ ABS( VL( K, KI+1 ) ) )
240 CONTINUE
REMAX = ONE / EMAX
CALL DSCAL( N, REMAX, VL( 1, KI ), 1 )
CALL DSCAL( N, REMAX, VL( 1, KI+1 ), 1 )
*
END IF
*
END IF
*
IS = IS + 1
IF( IP.NE.0 )
$ IS = IS + 1
250 CONTINUE
IF( IP.EQ.-1 )
$ IP = 0
IF( IP.EQ.1 )
$ IP = -1
*
260 CONTINUE
*
END IF
*
RETURN
*
* End of DTREVC
*
END
*> \brief \b DTREXC
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTREXC + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTREXC( COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPQ
* INTEGER IFST, ILST, INFO, LDQ, LDT, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTREXC reorders the real Schur factorization of a real matrix
*> A = Q*T*Q**T, so that the diagonal block of T with row index IFST is
*> moved to row ILST.
*>
*> The real Schur form T is reordered by an orthogonal similarity
*> transformation Z**T*T*Z, and optionally the matrix Q of Schur vectors
*> is updated by postmultiplying it with Z.
*>
*> T must be in Schur canonical form (as returned by DHSEQR), that is,
*> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
*> 2-by-2 diagonal block has its diagonal elements equal and its
*> off-diagonal elements of opposite sign.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] COMPQ
*> \verbatim
*> COMPQ is CHARACTER*1
*> = 'V': update the matrix Q of Schur vectors;
*> = 'N': do not update Q.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in,out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> On entry, the upper quasi-triangular matrix T, in Schur
*> Schur canonical form.
*> On exit, the reordered upper quasi-triangular matrix, again
*> in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
*> On exit, if COMPQ = 'V', Q has been postmultiplied by the
*> orthogonal transformation matrix Z which reorders T.
*> If COMPQ = 'N', Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] IFST
*> \verbatim
*> IFST is INTEGER
*> \endverbatim
*>
*> \param[in,out] ILST
*> \verbatim
*> ILST is INTEGER
*>
*> Specify the reordering of the diagonal blocks of T.
*> The block with row index IFST is moved to row ILST, by a
*> sequence of transpositions between adjacent blocks.
*> On exit, if IFST pointed on entry to the second row of a
*> 2-by-2 block, it is changed to point to the first row; ILST
*> always points to the first row of the block in its final
*> position (which may differ from its input value by +1 or -1).
*> 1 <= IFST <= N; 1 <= ILST <= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> = 1: two adjacent blocks were too close to swap (the problem
*> is very ill-conditioned); T may have been partially
*> reordered, and ILST points to the first row of the
*> current position of the block being moved.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DTREXC( COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK,
$ INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER COMPQ
INTEGER IFST, ILST, INFO, LDQ, LDT, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL WANTQ
INTEGER HERE, NBF, NBL, NBNEXT
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLAEXC, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Decode and test the input arguments.
*
INFO = 0
WANTQ = LSAME( COMPQ, 'V' )
IF( .NOT.WANTQ .AND. .NOT.LSAME( COMPQ, 'N' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.MAX( 1, N ) ) ) THEN
INFO = -6
ELSE IF( IFST.LT.1 .OR. IFST.GT.N ) THEN
INFO = -7
ELSE IF( ILST.LT.1 .OR. ILST.GT.N ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTREXC', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
* Determine the first row of specified block
* and find out it is 1 by 1 or 2 by 2.
*
IF( IFST.GT.1 ) THEN
IF( T( IFST, IFST-1 ).NE.ZERO )
$ IFST = IFST - 1
END IF
NBF = 1
IF( IFST.LT.N ) THEN
IF( T( IFST+1, IFST ).NE.ZERO )
$ NBF = 2
END IF
*
* Determine the first row of the final block
* and find out it is 1 by 1 or 2 by 2.
*
IF( ILST.GT.1 ) THEN
IF( T( ILST, ILST-1 ).NE.ZERO )
$ ILST = ILST - 1
END IF
NBL = 1
IF( ILST.LT.N ) THEN
IF( T( ILST+1, ILST ).NE.ZERO )
$ NBL = 2
END IF
*
IF( IFST.EQ.ILST )
$ RETURN
*
IF( IFST.LT.ILST ) THEN
*
* Update ILST
*
IF( NBF.EQ.2 .AND. NBL.EQ.1 )
$ ILST = ILST - 1
IF( NBF.EQ.1 .AND. NBL.EQ.2 )
$ ILST = ILST + 1
*
HERE = IFST
*
10 CONTINUE
*
* Swap block with next one below
*
IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN
*
* Current block either 1 by 1 or 2 by 2
*
NBNEXT = 1
IF( HERE+NBF+1.LE.N ) THEN
IF( T( HERE+NBF+1, HERE+NBF ).NE.ZERO )
$ NBNEXT = 2
END IF
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, NBF, NBNEXT,
$ WORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE + NBNEXT
*
* Test if 2 by 2 block breaks into two 1 by 1 blocks
*
IF( NBF.EQ.2 ) THEN
IF( T( HERE+1, HERE ).EQ.ZERO )
$ NBF = 3
END IF
*
ELSE
*
* Current block consists of two 1 by 1 blocks each of which
* must be swapped individually
*
NBNEXT = 1
IF( HERE+3.LE.N ) THEN
IF( T( HERE+3, HERE+2 ).NE.ZERO )
$ NBNEXT = 2
END IF
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE+1, 1, NBNEXT,
$ WORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
IF( NBNEXT.EQ.1 ) THEN
*
* Swap two 1 by 1 blocks, no problems possible
*
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1, NBNEXT,
$ WORK, INFO )
HERE = HERE + 1
ELSE
*
* Recompute NBNEXT in case 2 by 2 split
*
IF( T( HERE+2, HERE+1 ).EQ.ZERO )
$ NBNEXT = 1
IF( NBNEXT.EQ.2 ) THEN
*
* 2 by 2 Block did not split
*
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1,
$ NBNEXT, WORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE + 2
ELSE
*
* 2 by 2 Block did split
*
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1, 1,
$ WORK, INFO )
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE+1, 1, 1,
$ WORK, INFO )
HERE = HERE + 2
END IF
END IF
END IF
IF( HERE.LT.ILST )
$ GO TO 10
*
ELSE
*
HERE = IFST
20 CONTINUE
*
* Swap block with next one above
*
IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN
*
* Current block either 1 by 1 or 2 by 2
*
NBNEXT = 1
IF( HERE.GE.3 ) THEN
IF( T( HERE-1, HERE-2 ).NE.ZERO )
$ NBNEXT = 2
END IF
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-NBNEXT, NBNEXT,
$ NBF, WORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE - NBNEXT
*
* Test if 2 by 2 block breaks into two 1 by 1 blocks
*
IF( NBF.EQ.2 ) THEN
IF( T( HERE+1, HERE ).EQ.ZERO )
$ NBF = 3
END IF
*
ELSE
*
* Current block consists of two 1 by 1 blocks each of which
* must be swapped individually
*
NBNEXT = 1
IF( HERE.GE.3 ) THEN
IF( T( HERE-1, HERE-2 ).NE.ZERO )
$ NBNEXT = 2
END IF
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-NBNEXT, NBNEXT,
$ 1, WORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
IF( NBNEXT.EQ.1 ) THEN
*
* Swap two 1 by 1 blocks, no problems possible
*
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, NBNEXT, 1,
$ WORK, INFO )
HERE = HERE - 1
ELSE
*
* Recompute NBNEXT in case 2 by 2 split
*
IF( T( HERE, HERE-1 ).EQ.ZERO )
$ NBNEXT = 1
IF( NBNEXT.EQ.2 ) THEN
*
* 2 by 2 Block did not split
*
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-1, 2, 1,
$ WORK, INFO )
IF( INFO.NE.0 ) THEN
ILST = HERE
RETURN
END IF
HERE = HERE - 2
ELSE
*
* 2 by 2 Block did split
*
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1, 1,
$ WORK, INFO )
CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-1, 1, 1,
$ WORK, INFO )
HERE = HERE - 2
END IF
END IF
END IF
IF( HERE.GT.ILST )
$ GO TO 20
END IF
ILST = HERE
*
RETURN
*
* End of DTREXC
*
END
*> \brief \b IEEECK
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download IEEECK + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION IEEECK( ISPEC, ZERO, ONE )
*
* .. Scalar Arguments ..
* INTEGER ISPEC
* REAL ONE, ZERO
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> IEEECK is called from the ILAENV to verify that Infinity and
*> possibly NaN arithmetic is safe (i.e. will not trap).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ISPEC
*> \verbatim
*> ISPEC is INTEGER
*> Specifies whether to test just for inifinity arithmetic
*> or whether to test for infinity and NaN arithmetic.
*> = 0: Verify infinity arithmetic only.
*> = 1: Verify infinity and NaN arithmetic.
*> \endverbatim
*>
*> \param[in] ZERO
*> \verbatim
*> ZERO is REAL
*> Must contain the value 0.0
*> This is passed to prevent the compiler from optimizing
*> away this code.
*> \endverbatim
*>
*> \param[in] ONE
*> \verbatim
*> ONE is REAL
*> Must contain the value 1.0
*> This is passed to prevent the compiler from optimizing
*> away this code.
*>
*> RETURN VALUE: INTEGER
*> = 0: Arithmetic failed to produce the correct answers
*> = 1: Arithmetic produced the correct answers
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
INTEGER FUNCTION IEEECK( ISPEC, ZERO, ONE )
*
* -- LAPACK auxiliary routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER ISPEC
REAL ONE, ZERO
* ..
*
* =====================================================================
*
* .. Local Scalars ..
REAL NAN1, NAN2, NAN3, NAN4, NAN5, NAN6, NEGINF,
$ NEGZRO, NEWZRO, POSINF
* ..
* .. Executable Statements ..
IEEECK = 1
*
POSINF = ONE / ZERO
IF( POSINF.LE.ONE ) THEN
IEEECK = 0
RETURN
END IF
*
NEGINF = -ONE / ZERO
IF( NEGINF.GE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
NEGZRO = ONE / ( NEGINF+ONE )
IF( NEGZRO.NE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
NEGINF = ONE / NEGZRO
IF( NEGINF.GE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
NEWZRO = NEGZRO + ZERO
IF( NEWZRO.NE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
POSINF = ONE / NEWZRO
IF( POSINF.LE.ONE ) THEN
IEEECK = 0
RETURN
END IF
*
NEGINF = NEGINF*POSINF
IF( NEGINF.GE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
POSINF = POSINF*POSINF
IF( POSINF.LE.ONE ) THEN
IEEECK = 0
RETURN
END IF
*
*
*
*
* Return if we were only asked to check infinity arithmetic
*
IF( ISPEC.EQ.0 )
$ RETURN
*
NAN1 = POSINF + NEGINF
*
NAN2 = POSINF / NEGINF
*
NAN3 = POSINF / POSINF
*
NAN4 = POSINF*ZERO
*
NAN5 = NEGINF*NEGZRO
*
NAN6 = NAN5*ZERO
*
IF( NAN1.EQ.NAN1 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN2.EQ.NAN2 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN3.EQ.NAN3 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN4.EQ.NAN4 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN5.EQ.NAN5 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN6.EQ.NAN6 ) THEN
IEEECK = 0
RETURN
END IF
*
RETURN
END
*> \brief \b ILADLC scans a matrix for its last non-zero column.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILADLC + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION ILADLC( M, N, A, LDA )
*
* .. Scalar Arguments ..
* INTEGER M, N, LDA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ILADLC scans A for its last non-zero column.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The m by n matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
INTEGER FUNCTION ILADLC( M, N, A, LDA )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER M, N, LDA
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
* ..
* .. Executable Statements ..
*
* Quick test for the common case where one corner is non-zero.
IF( N.EQ.0 ) THEN
ILADLC = N
ELSE IF( A(1, N).NE.ZERO .OR. A(M, N).NE.ZERO ) THEN
ILADLC = N
ELSE
* Now scan each column from the end, returning with the first non-zero.
DO ILADLC = N, 1, -1
DO I = 1, M
IF( A(I, ILADLC).NE.ZERO ) RETURN
END DO
END DO
END IF
RETURN
END
*> \brief \b ILADLR scans a matrix for its last non-zero row.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILADLR + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION ILADLR( M, N, A, LDA )
*
* .. Scalar Arguments ..
* INTEGER M, N, LDA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ILADLR scans A for its last non-zero row.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The m by n matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
INTEGER FUNCTION ILADLR( M, N, A, LDA )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER M, N, LDA
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
* ..
* .. Executable Statements ..
*
* Quick test for the common case where one corner is non-zero.
IF( M.EQ.0 ) THEN
ILADLR = M
ELSE IF( A(M, 1).NE.ZERO .OR. A(M, N).NE.ZERO ) THEN
ILADLR = M
ELSE
* Scan up each column tracking the last zero row seen.
ILADLR = 0
DO J = 1, N
I=M
DO WHILE((A(MAX(I,1),J).EQ.ZERO).AND.(I.GE.1))
I=I-1
ENDDO
ILADLR = MAX( ILADLR, I )
END DO
END IF
RETURN
END
*> \brief \b ILAENV
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILAENV + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, N4 )
*
* .. Scalar Arguments ..
* CHARACTER*( * ) NAME, OPTS
* INTEGER ISPEC, N1, N2, N3, N4
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ILAENV is called from the LAPACK routines to choose problem-dependent
*> parameters for the local environment. See ISPEC for a description of
*> the parameters.
*>
*> ILAENV returns an INTEGER
*> if ILAENV >= 0: ILAENV returns the value of the parameter specified by ISPEC
*> if ILAENV < 0: if ILAENV = -k, the k-th argument had an illegal value.
*>
*> This version provides a set of parameters which should give good,
*> but not optimal, performance on many of the currently available
*> computers. Users are encouraged to modify this subroutine to set
*> the tuning parameters for their particular machine using the option
*> and problem size information in the arguments.
*>
*> This routine will not function correctly if it is converted to all
*> lower case. Converting it to all upper case is allowed.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ISPEC
*> \verbatim
*> ISPEC is INTEGER
*> Specifies the parameter to be returned as the value of
*> ILAENV.
*> = 1: the optimal blocksize; if this value is 1, an unblocked
*> algorithm will give the best performance.
*> = 2: the minimum block size for which the block routine
*> should be used; if the usable block size is less than
*> this value, an unblocked routine should be used.
*> = 3: the crossover point (in a block routine, for N less
*> than this value, an unblocked routine should be used)
*> = 4: the number of shifts, used in the nonsymmetric
*> eigenvalue routines (DEPRECATED)
*> = 5: the minimum column dimension for blocking to be used;
*> rectangular blocks must have dimension at least k by m,
*> where k is given by ILAENV(2,...) and m by ILAENV(5,...)
*> = 6: the crossover point for the SVD (when reducing an m by n
*> matrix to bidiagonal form, if max(m,n)/min(m,n) exceeds
*> this value, a QR factorization is used first to reduce
*> the matrix to a triangular form.)
*> = 7: the number of processors
*> = 8: the crossover point for the multishift QR method
*> for nonsymmetric eigenvalue problems (DEPRECATED)
*> = 9: maximum size of the subproblems at the bottom of the
*> computation tree in the divide-and-conquer algorithm
*> (used by xGELSD and xGESDD)
*> =10: ieee NaN arithmetic can be trusted not to trap
*> =11: infinity arithmetic can be trusted not to trap
*> 12 <= ISPEC <= 16:
*> xHSEQR or related subroutines,
*> see IPARMQ for detailed explanation
*> \endverbatim
*>
*> \param[in] NAME
*> \verbatim
*> NAME is CHARACTER*(*)
*> The name of the calling subroutine, in either upper case or
*> lower case.
*> \endverbatim
*>
*> \param[in] OPTS
*> \verbatim
*> OPTS is CHARACTER*(*)
*> The character options to the subroutine NAME, concatenated
*> into a single character string. For example, UPLO = 'U',
*> TRANS = 'T', and DIAG = 'N' for a triangular routine would
*> be specified as OPTS = 'UTN'.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*> N2 is INTEGER
*> \endverbatim
*>
*> \param[in] N3
*> \verbatim
*> N3 is INTEGER
*> \endverbatim
*>
*> \param[in] N4
*> \verbatim
*> N4 is INTEGER
*> Problem dimensions for the subroutine NAME; these may not all
*> be required.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup auxOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The following conventions have been used when calling ILAENV from the
*> LAPACK routines:
*> 1) OPTS is a concatenation of all of the character options to
*> subroutine NAME, in the same order that they appear in the
*> argument list for NAME, even if they are not used in determining
*> the value of the parameter specified by ISPEC.
*> 2) The problem dimensions N1, N2, N3, N4 are specified in the order
*> that they appear in the argument list for NAME. N1 is used
*> first, N2 second, and so on, and unused problem dimensions are
*> passed a value of -1.
*> 3) The parameter value returned by ILAENV is checked for validity in
*> the calling subroutine. For example, ILAENV is used to retrieve
*> the optimal blocksize for STRTRI as follows:
*>
*> NB = ILAENV( 1, 'STRTRI', UPLO // DIAG, N, -1, -1, -1 )
*> IF( NB.LE.1 ) NB = MAX( 1, N )
*> \endverbatim
*>
* =====================================================================
INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, N4 )
*
* -- LAPACK auxiliary routine (version 3.6.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
CHARACTER*( * ) NAME, OPTS
INTEGER ISPEC, N1, N2, N3, N4
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, IC, IZ, NB, NBMIN, NX
LOGICAL CNAME, SNAME
CHARACTER C1*1, C2*2, C4*2, C3*3, SUBNAM*6
* ..
* .. Intrinsic Functions ..
INTRINSIC CHAR, ICHAR, INT, MIN, REAL
* ..
* .. External Functions ..
INTEGER IEEECK, IPARMQ
EXTERNAL IEEECK, IPARMQ
* ..
* .. Executable Statements ..
*
GO TO ( 10, 10, 10, 80, 90, 100, 110, 120,
$ 130, 140, 150, 160, 160, 160, 160, 160 )ISPEC
*
* Invalid value for ISPEC
*
ILAENV = -1
RETURN
*
10 CONTINUE
*
* Convert NAME to upper case if the first character is lower case.
*
ILAENV = 1
SUBNAM = NAME
IC = ICHAR( SUBNAM( 1: 1 ) )
IZ = ICHAR( 'Z' )
IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN
*
* ASCII character set
*
IF( IC.GE.97 .AND. IC.LE.122 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO 20 I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.97 .AND. IC.LE.122 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
20 CONTINUE
END IF
*
ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN
*
* EBCDIC character set
*
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) ) THEN
SUBNAM( 1: 1 ) = CHAR( IC+64 )
DO 30 I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) )SUBNAM( I:
$ I ) = CHAR( IC+64 )
30 CONTINUE
END IF
*
ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN
*
* Prime machines: ASCII+128
*
IF( IC.GE.225 .AND. IC.LE.250 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO 40 I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.225 .AND. IC.LE.250 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
40 CONTINUE
END IF
END IF
*
C1 = SUBNAM( 1: 1 )
SNAME = C1.EQ.'S' .OR. C1.EQ.'D'
CNAME = C1.EQ.'C' .OR. C1.EQ.'Z'
IF( .NOT.( CNAME .OR. SNAME ) )
$ RETURN
C2 = SUBNAM( 2: 3 )
C3 = SUBNAM( 4: 6 )
C4 = C3( 2: 3 )
*
GO TO ( 50, 60, 70 )ISPEC
*
50 CONTINUE
*
* ISPEC = 1: block size
*
* In these examples, separate code is provided for setting NB for
* real and complex. We assume that NB will take the same value in
* single or double precision.
*
NB = 1
*
IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR.
$ C3.EQ.'QLF' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
END IF
ELSE IF( C2.EQ.'PO' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NB = 32
ELSE IF( SNAME .AND. C3.EQ.'GST' ) THEN
NB = 64
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRF' ) THEN
NB = 64
ELSE IF( C3.EQ.'TRD' ) THEN
NB = 32
ELSE IF( C3.EQ.'GST' ) THEN
NB = 64
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
END IF
ELSE IF( C2.EQ.'GB' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
IF( N4.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
ELSE
IF( N4.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
END IF
END IF
ELSE IF( C2.EQ.'PB' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
IF( N2.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
ELSE
IF( N2.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
END IF
END IF
ELSE IF( C2.EQ.'TR' ) THEN
IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF ( C3.EQ.'EVC' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
END IF
ELSE IF( C2.EQ.'LA' ) THEN
IF( C3.EQ.'UUM' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
END IF
ELSE IF( SNAME .AND. C2.EQ.'ST' ) THEN
IF( C3.EQ.'EBZ' ) THEN
NB = 1
END IF
ELSE IF( C2.EQ.'GG' ) THEN
NB = 32
IF( C3.EQ.'HD3' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
END IF
END IF
ILAENV = NB
RETURN
*
60 CONTINUE
*
* ISPEC = 2: minimum block size
*
NBMIN = 2
IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. C3.EQ.
$ 'QLF' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NBMIN = 8
ELSE
NBMIN = 8
END IF
ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NBMIN = 2
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRD' ) THEN
NBMIN = 2
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
END IF
ELSE IF( C2.EQ.'GG' ) THEN
NBMIN = 2
IF( C3.EQ.'HD3' ) THEN
NBMIN = 2
END IF
END IF
ILAENV = NBMIN
RETURN
*
70 CONTINUE
*
* ISPEC = 3: crossover point
*
NX = 0
IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. C3.EQ.
$ 'QLF' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NX = 32
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRD' ) THEN
NX = 32
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NX = 128
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NX = 128
END IF
END IF
ELSE IF( C2.EQ.'GG' ) THEN
NX = 128
IF( C3.EQ.'HD3' ) THEN
NX = 128
END IF
END IF
ILAENV = NX
RETURN
*
80 CONTINUE
*
* ISPEC = 4: number of shifts (used by xHSEQR)
*
ILAENV = 6
RETURN
*
90 CONTINUE
*
* ISPEC = 5: minimum column dimension (not used)
*
ILAENV = 2
RETURN
*
100 CONTINUE
*
* ISPEC = 6: crossover point for SVD (used by xGELSS and xGESVD)
*
ILAENV = INT( REAL( MIN( N1, N2 ) )*1.6E0 )
RETURN
*
110 CONTINUE
*
* ISPEC = 7: number of processors (not used)
*
ILAENV = 1
RETURN
*
120 CONTINUE
*
* ISPEC = 8: crossover point for multishift (used by xHSEQR)
*
ILAENV = 50
RETURN
*
130 CONTINUE
*
* ISPEC = 9: maximum size of the subproblems at the bottom of the
* computation tree in the divide-and-conquer algorithm
* (used by xGELSD and xGESDD)
*
ILAENV = 25
RETURN
*
140 CONTINUE
*
* ISPEC = 10: ieee NaN arithmetic can be trusted not to trap
*
* ILAENV = 0
ILAENV = 1
IF( ILAENV.EQ.1 ) THEN
ILAENV = IEEECK( 1, 0.0, 1.0 )
END IF
RETURN
*
150 CONTINUE
*
* ISPEC = 11: infinity arithmetic can be trusted not to trap
*
* ILAENV = 0
ILAENV = 1
IF( ILAENV.EQ.1 ) THEN
ILAENV = IEEECK( 0, 0.0, 1.0 )
END IF
RETURN
*
160 CONTINUE
*
* 12 <= ISPEC <= 16: xHSEQR or related subroutines.
*
ILAENV = IPARMQ( ISPEC, NAME, OPTS, N1, N2, N3, N4 )
RETURN
*
* End of ILAENV
*
END
*> \brief \b IPARMQ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download IPARMQ + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION IPARMQ( ISPEC, NAME, OPTS, N, ILO, IHI, LWORK )
*
* .. Scalar Arguments ..
* INTEGER IHI, ILO, ISPEC, LWORK, N
* CHARACTER NAME*( * ), OPTS*( * )
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This program sets problem and machine dependent parameters
*> useful for xHSEQR and related subroutines for eigenvalue
*> problems. It is called whenever
*> IPARMQ is called with 12 <= ISPEC <= 16
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ISPEC
*> \verbatim
*> ISPEC is integer scalar
*> ISPEC specifies which tunable parameter IPARMQ should
*> return.
*>
*> ISPEC=12: (INMIN) Matrices of order nmin or less
*> are sent directly to xLAHQR, the implicit
*> double shift QR algorithm. NMIN must be
*> at least 11.
*>
*> ISPEC=13: (INWIN) Size of the deflation window.
*> This is best set greater than or equal to
*> the number of simultaneous shifts NS.
*> Larger matrices benefit from larger deflation
*> windows.
*>
*> ISPEC=14: (INIBL) Determines when to stop nibbling and
*> invest in an (expensive) multi-shift QR sweep.
*> If the aggressive early deflation subroutine
*> finds LD converged eigenvalues from an order
*> NW deflation window and LD.GT.(NW*NIBBLE)/100,
*> then the next QR sweep is skipped and early
*> deflation is applied immediately to the
*> remaining active diagonal block. Setting
*> IPARMQ(ISPEC=14) = 0 causes TTQRE to skip a
*> multi-shift QR sweep whenever early deflation
*> finds a converged eigenvalue. Setting
*> IPARMQ(ISPEC=14) greater than or equal to 100
*> prevents TTQRE from skipping a multi-shift
*> QR sweep.
*>
*> ISPEC=15: (NSHFTS) The number of simultaneous shifts in
*> a multi-shift QR iteration.
*>
*> ISPEC=16: (IACC22) IPARMQ is set to 0, 1 or 2 with the
*> following meanings.
*> 0: During the multi-shift QR/QZ sweep,
*> blocked eigenvalue reordering, blocked
*> Hessenberg-triangular reduction,
*> reflections and/or rotations are not
*> accumulated when updating the
*> far-from-diagonal matrix entries.
*> 1: During the multi-shift QR/QZ sweep,
*> blocked eigenvalue reordering, blocked
*> Hessenberg-triangular reduction,
*> reflections and/or rotations are
*> accumulated, and matrix-matrix
*> multiplication is used to update the
*> far-from-diagonal matrix entries.
*> 2: During the multi-shift QR/QZ sweep,
*> blocked eigenvalue reordering, blocked
*> Hessenberg-triangular reduction,
*> reflections and/or rotations are
*> accumulated, and 2-by-2 block structure
*> is exploited during matrix-matrix
*> multiplies.
*> (If xTRMM is slower than xGEMM, then
*> IPARMQ(ISPEC=16)=1 may be more efficient than
*> IPARMQ(ISPEC=16)=2 despite the greater level of
*> arithmetic work implied by the latter choice.)
*> \endverbatim
*>
*> \param[in] NAME
*> \verbatim
*> NAME is character string
*> Name of the calling subroutine
*> \endverbatim
*>
*> \param[in] OPTS
*> \verbatim
*> OPTS is character string
*> This is a concatenation of the string arguments to
*> TTQRE.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is integer scalar
*> N is the order of the Hessenberg matrix H.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> It is assumed that H is already upper triangular
*> in rows and columns 1:ILO-1 and IHI+1:N.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is integer scalar
*> The amount of workspace available.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2015
*
*> \ingroup auxOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Little is known about how best to choose these parameters.
*> It is possible to use different values of the parameters
*> for each of CHSEQR, DHSEQR, SHSEQR and ZHSEQR.
*>
*> It is probably best to choose different parameters for
*> different matrices and different parameters at different
*> times during the iteration, but this has not been
*> implemented --- yet.
*>
*>
*> The best choices of most of the parameters depend
*> in an ill-understood way on the relative execution
*> rate of xLAQR3 and xLAQR5 and on the nature of each
*> particular eigenvalue problem. Experiment may be the
*> only practical way to determine which choices are most
*> effective.
*>
*> Following is a list of default values supplied by IPARMQ.
*> These defaults may be adjusted in order to attain better
*> performance in any particular computational environment.
*>
*> IPARMQ(ISPEC=12) The xLAHQR vs xLAQR0 crossover point.
*> Default: 75. (Must be at least 11.)
*>
*> IPARMQ(ISPEC=13) Recommended deflation window size.
*> This depends on ILO, IHI and NS, the
*> number of simultaneous shifts returned
*> by IPARMQ(ISPEC=15). The default for
*> (IHI-ILO+1).LE.500 is NS. The default
*> for (IHI-ILO+1).GT.500 is 3*NS/2.
*>
*> IPARMQ(ISPEC=14) Nibble crossover point. Default: 14.
*>
*> IPARMQ(ISPEC=15) Number of simultaneous shifts, NS.
*> a multi-shift QR iteration.
*>
*> If IHI-ILO+1 is ...
*>
*> greater than ...but less ... the
*> or equal to ... than default is
*>
*> 0 30 NS = 2+
*> 30 60 NS = 4+
*> 60 150 NS = 10
*> 150 590 NS = **
*> 590 3000 NS = 64
*> 3000 6000 NS = 128
*> 6000 infinity NS = 256
*>
*> (+) By default matrices of this order are
*> passed to the implicit double shift routine
*> xLAHQR. See IPARMQ(ISPEC=12) above. These
*> values of NS are used only in case of a rare
*> xLAHQR failure.
*>
*> (**) The asterisks (**) indicate an ad-hoc
*> function increasing from 10 to 64.
*>
*> IPARMQ(ISPEC=16) Select structured matrix multiply.
*> (See ISPEC=16 above for details.)
*> Default: 3.
*> \endverbatim
*>
* =====================================================================
INTEGER FUNCTION IPARMQ( ISPEC, NAME, OPTS, N, ILO, IHI, LWORK )
*
* -- LAPACK auxiliary routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2015
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, ISPEC, LWORK, N
CHARACTER NAME*( * ), OPTS*( * )
*
* ================================================================
* .. Parameters ..
INTEGER INMIN, INWIN, INIBL, ISHFTS, IACC22
PARAMETER ( INMIN = 12, INWIN = 13, INIBL = 14,
$ ISHFTS = 15, IACC22 = 16 )
INTEGER NMIN, K22MIN, KACMIN, NIBBLE, KNWSWP
PARAMETER ( NMIN = 75, K22MIN = 14, KACMIN = 14,
$ NIBBLE = 14, KNWSWP = 500 )
REAL TWO
PARAMETER ( TWO = 2.0 )
* ..
* .. Local Scalars ..
INTEGER NH, NS
INTEGER I, IC, IZ
CHARACTER SUBNAM*6
* ..
* .. Intrinsic Functions ..
INTRINSIC LOG, MAX, MOD, NINT, REAL
* ..
* .. Executable Statements ..
IF( ( ISPEC.EQ.ISHFTS ) .OR. ( ISPEC.EQ.INWIN ) .OR.
$ ( ISPEC.EQ.IACC22 ) ) THEN
*
* ==== Set the number simultaneous shifts ====
*
NH = IHI - ILO + 1
NS = 2
IF( NH.GE.30 )
$ NS = 4
IF( NH.GE.60 )
$ NS = 10
IF( NH.GE.150 )
$ NS = MAX( 10, NH / NINT( LOG( REAL( NH ) ) / LOG( TWO ) ) )
IF( NH.GE.590 )
$ NS = 64
IF( NH.GE.3000 )
$ NS = 128
IF( NH.GE.6000 )
$ NS = 256
NS = MAX( 2, NS-MOD( NS, 2 ) )
END IF
*
IF( ISPEC.EQ.INMIN ) THEN
*
*
* ===== Matrices of order smaller than NMIN get sent
* . to xLAHQR, the classic double shift algorithm.
* . This must be at least 11. ====
*
IPARMQ = NMIN
*
ELSE IF( ISPEC.EQ.INIBL ) THEN
*
* ==== INIBL: skip a multi-shift qr iteration and
* . whenever aggressive early deflation finds
* . at least (NIBBLE*(window size)/100) deflations. ====
*
IPARMQ = NIBBLE
*
ELSE IF( ISPEC.EQ.ISHFTS ) THEN
*
* ==== NSHFTS: The number of simultaneous shifts =====
*
IPARMQ = NS
*
ELSE IF( ISPEC.EQ.INWIN ) THEN
*
* ==== NW: deflation window size. ====
*
IF( NH.LE.KNWSWP ) THEN
IPARMQ = NS
ELSE
IPARMQ = 3*NS / 2
END IF
*
ELSE IF( ISPEC.EQ.IACC22 ) THEN
*
* ==== IACC22: Whether to accumulate reflections
* . before updating the far-from-diagonal elements
* . and whether to use 2-by-2 block structure while
* . doing it. A small amount of work could be saved
* . by making this choice dependent also upon the
* . NH=IHI-ILO+1.
*
*
* Convert NAME to upper case if the first character is lower case.
*
IPARMQ = 0
SUBNAM = NAME
IC = ICHAR( SUBNAM( 1: 1 ) )
IZ = ICHAR( 'Z' )
IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN
*
* ASCII character set
*
IF( IC.GE.97 .AND. IC.LE.122 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.97 .AND. IC.LE.122 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
END DO
END IF
*
ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN
*
* EBCDIC character set
*
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) ) THEN
SUBNAM( 1: 1 ) = CHAR( IC+64 )
DO I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) )SUBNAM( I:
$ I ) = CHAR( IC+64 )
END DO
END IF
*
ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN
*
* Prime machines: ASCII+128
*
IF( IC.GE.225 .AND. IC.LE.250 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.225 .AND. IC.LE.250 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
END DO
END IF
END IF
*
IF( SUBNAM( 2:6 ).EQ.'GGHRD' .OR.
$ SUBNAM( 2:6 ).EQ.'GGHD3' ) THEN
IPARMQ = 1
IF( NH.GE.K22MIN )
$ IPARMQ = 2
ELSE IF ( SUBNAM( 4:6 ).EQ.'EXC' ) THEN
IF( NH.GE.KACMIN )
$ IPARMQ = 1
IF( NH.GE.K22MIN )
$ IPARMQ = 2
ELSE IF ( SUBNAM( 2:6 ).EQ.'HSEQR' .OR.
$ SUBNAM( 2:5 ).EQ.'LAQR' ) THEN
IF( NS.GE.KACMIN )
$ IPARMQ = 1
IF( NS.GE.K22MIN )
$ IPARMQ = 2
END IF
*
ELSE
* ===== invalid value of ispec =====
IPARMQ = -1
*
END IF
*
* ==== End of IPARMQ ====
*
END
*> \brief \b LSAME
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* LOGICAL FUNCTION LSAME( CA, CB )
*
* .. Scalar Arguments ..
* CHARACTER CA, CB
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> LSAME returns .TRUE. if CA is the same letter as CB regardless of
*> case.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] CA
*> \verbatim
*> \endverbatim
*>
*> \param[in] CB
*> \verbatim
*> CA and CB specify the single characters to be compared.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
LOGICAL FUNCTION LSAME( CA, CB )
*
* -- LAPACK auxiliary routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER CA, CB
* ..
*
* =====================================================================
*
* .. Intrinsic Functions ..
INTRINSIC ICHAR
* ..
* .. Local Scalars ..
INTEGER INTA, INTB, ZCODE
* ..
* .. Executable Statements ..
*
* Test if the characters are equal
*
LSAME = CA.EQ.CB
IF( LSAME )
$ RETURN
*
* Now test for equivalence if both characters are alphabetic.
*
ZCODE = ICHAR( 'Z' )
*
* Use 'Z' rather than 'A' so that ASCII can be detected on Prime
* machines, on which ICHAR returns a value with bit 8 set.
* ICHAR('A') on Prime machines returns 193 which is the same as
* ICHAR('A') on an EBCDIC machine.
*
INTA = ICHAR( CA )
INTB = ICHAR( CB )
*
IF( ZCODE.EQ.90 .OR. ZCODE.EQ.122 ) THEN
*
* ASCII is assumed - ZCODE is the ASCII code of either lower or
* upper case 'Z'.
*
IF( INTA.GE.97 .AND. INTA.LE.122 ) INTA = INTA - 32
IF( INTB.GE.97 .AND. INTB.LE.122 ) INTB = INTB - 32
*
ELSE IF( ZCODE.EQ.233 .OR. ZCODE.EQ.169 ) THEN
*
* EBCDIC is assumed - ZCODE is the EBCDIC code of either lower or
* upper case 'Z'.
*
IF( INTA.GE.129 .AND. INTA.LE.137 .OR.
$ INTA.GE.145 .AND. INTA.LE.153 .OR.
$ INTA.GE.162 .AND. INTA.LE.169 ) INTA = INTA + 64
IF( INTB.GE.129 .AND. INTB.LE.137 .OR.
$ INTB.GE.145 .AND. INTB.LE.153 .OR.
$ INTB.GE.162 .AND. INTB.LE.169 ) INTB = INTB + 64
*
ELSE IF( ZCODE.EQ.218 .OR. ZCODE.EQ.250 ) THEN
*
* ASCII is assumed, on Prime machines - ZCODE is the ASCII code
* plus 128 of either lower or upper case 'Z'.
*
IF( INTA.GE.225 .AND. INTA.LE.250 ) INTA = INTA - 32
IF( INTB.GE.225 .AND. INTB.LE.250 ) INTB = INTB - 32
END IF
LSAME = INTA.EQ.INTB
*
* RETURN
*
* End of LSAME
*
END
*> \brief \b XERBLA
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download XERBLA + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE XERBLA( SRNAME, INFO )
*
* .. Scalar Arguments ..
* CHARACTER*(*) SRNAME
* INTEGER INFO
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> XERBLA is an error handler for the LAPACK routines.
*> It is called by an LAPACK routine if an input parameter has an
*> invalid value. A message is printed and execution stops.
*>
*> Installers may consider modifying the STOP statement in order to
*> call system-specific exception-handling facilities.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SRNAME
*> \verbatim
*> SRNAME is CHARACTER*(*)
*> The name of the routine which called XERBLA.
*> \endverbatim
*>
*> \param[in] INFO
*> \verbatim
*> INFO is INTEGER
*> The position of the invalid parameter in the parameter list
*> of the calling routine.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
SUBROUTINE XERBLA( SRNAME, INFO )
*
* -- LAPACK auxiliary routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER*(*) SRNAME
INTEGER INFO
* ..
*
* =====================================================================
*
* .. Intrinsic Functions ..
INTRINSIC LEN_TRIM
* ..
* .. Executable Statements ..
*
WRITE( *, FMT = 9999 )SRNAME( 1:LEN_TRIM( SRNAME ) ), INFO
*
STOP
*
9999 FORMAT( ' ** On entry to ', A, ' parameter number ', I2, ' had ',
$ 'an illegal value' )
*
* End of XERBLA
*
END