514 lines
18 KiB
Fortran
514 lines
18 KiB
Fortran
*> \brief \b SHSEQR
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download SHSEQR + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/shseqr.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/shseqr.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/shseqr.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE SHSEQR( 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 ..
|
|
* REAL H( LDH, * ), WI( * ), WORK( * ), WR( * ),
|
|
* $ Z( LDZ, * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> SHSEQR 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 >= 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 SGEBAL, and then passed to ZGEHRD
|
|
*> when the matrix output by SGEBAL is reduced to Hessenberg
|
|
*> form. Otherwise ILO and IHI should be set to 1 and N
|
|
*> respectively. If N > 0, then 1 <= ILO <= IHI <= N.
|
|
*> If N = 0, then ILO = 1 and IHI = 0.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] H
|
|
*> \verbatim
|
|
*> H is REAL 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) < 0. If INFO = 0 and JOB = 'E', the
|
|
*> contents of H are unspecified on exit. (The output value of
|
|
*> H when INFO > 0 is given under the description of INFO
|
|
*> below.)
|
|
*>
|
|
*> Unlike earlier versions of SHSEQR, this subroutine may
|
|
*> explicitly H(i,j) = 0 for i > 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 >= max(1,N).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] WR
|
|
*> \verbatim
|
|
*> WR is REAL array, dimension (N)
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] WI
|
|
*> \verbatim
|
|
*> WI is REAL 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) > 0 and
|
|
*> WI(i+1) < 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 REAL 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 SORGHR
|
|
*> after the call to SGEHRD which formed the Hessenberg matrix
|
|
*> H. (The output value of Z when INFO > 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 >= MAX(1,N). Otherwise, LDZ >= 1.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] WORK
|
|
*> \verbatim
|
|
*> WORK is REAL 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 >= 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 SHSEQR does a workspace query.
|
|
*> In this case, SHSEQR 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
|
|
*> < 0: if INFO = -i, the i-th argument had an illegal
|
|
*> value
|
|
*> > 0: if INFO = i, SHSEQR 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 > 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 > 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 > 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 > 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 > 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.
|
|
*
|
|
*> \ingroup realOTHERcomputational
|
|
*
|
|
*> \par Contributors:
|
|
* ==================
|
|
*>
|
|
*> Karen Braman and Ralph Byers, Department of Mathematics,
|
|
*> University of Kansas, USA
|
|
*
|
|
*> \par Further Details:
|
|
* =====================
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> Default values supplied by
|
|
*> ILAENV(ISPEC,'SHSEQR',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 SLAHQR vs SLAQR0 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) <= 500 is NS.
|
|
*> The default for (IHI-ILO+1) > 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
|
|
*> SLAHQR 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 SHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
|
|
$ LDZ, WORK, LWORK, INFO )
|
|
*
|
|
* -- LAPACK computational routine --
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
|
*
|
|
* .. Scalar Arguments ..
|
|
INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
|
|
CHARACTER COMPZ, JOB
|
|
* ..
|
|
* .. Array Arguments ..
|
|
REAL H( LDH, * ), WI( * ), WORK( * ), WR( * ),
|
|
$ Z( LDZ, * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
*
|
|
* ==== Matrices of order NTINY or smaller must be processed by
|
|
* . SLAHQR because of insufficient subdiagonal scratch space.
|
|
* . (This is a hard limit.) ====
|
|
INTEGER NTINY
|
|
PARAMETER ( NTINY = 15 )
|
|
*
|
|
* ==== NL allocates some local workspace to help small matrices
|
|
* . through a rare SLAHQR failure. NL > NTINY = 15 is
|
|
* . required and NL <= 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 )
|
|
REAL ZERO, ONE
|
|
PARAMETER ( ZERO = 0.0e0, ONE = 1.0e0 )
|
|
* ..
|
|
* .. Local Arrays ..
|
|
REAL 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 SLACPY, SLAHQR, SLAQR0, SLASET, XERBLA
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC MAX, MIN, REAL
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
* ==== Decode and check the input parameters. ====
|
|
*
|
|
WANTT = LSAME( JOB, 'S' )
|
|
INITZ = LSAME( COMPZ, 'I' )
|
|
WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
|
|
WORK( 1 ) = REAL( 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( 'SHSEQR', -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 SLAQR0( 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( REAL( MAX( 1, N ) ), WORK( 1 ) )
|
|
RETURN
|
|
*
|
|
ELSE
|
|
*
|
|
* ==== copy eigenvalues isolated by SGEBAL ====
|
|
*
|
|
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 SLASET( '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
|
|
*
|
|
* ==== SLAHQR/SLAQR0 crossover point ====
|
|
*
|
|
NMIN = ILAENV( 12, 'SHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N,
|
|
$ ILO, IHI, LWORK )
|
|
NMIN = MAX( NTINY, NMIN )
|
|
*
|
|
* ==== SLAQR0 for big matrices; SLAHQR for small ones ====
|
|
*
|
|
IF( N.GT.NMIN ) THEN
|
|
CALL SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
|
|
$ IHI, Z, LDZ, WORK, LWORK, INFO )
|
|
ELSE
|
|
*
|
|
* ==== Small matrix ====
|
|
*
|
|
CALL SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
|
|
$ IHI, Z, LDZ, INFO )
|
|
*
|
|
IF( INFO.GT.0 ) THEN
|
|
*
|
|
* ==== A rare SLAHQR failure! SLAQR0 sometimes succeeds
|
|
* . when SLAHQR fails. ====
|
|
*
|
|
KBOT = INFO
|
|
*
|
|
IF( N.GE.NL ) THEN
|
|
*
|
|
* ==== Larger matrices have enough subdiagonal scratch
|
|
* . space to call SLAQR0 directly. ====
|
|
*
|
|
CALL SLAQR0( 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 SLAQR0. Hence,
|
|
* . tiny matrices must be copied into a larger
|
|
* . array before calling SLAQR0. ====
|
|
*
|
|
CALL SLACPY( 'A', N, N, H, LDH, HL, NL )
|
|
HL( N+1, N ) = ZERO
|
|
CALL SLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ),
|
|
$ NL )
|
|
CALL SLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, WR,
|
|
$ WI, ILO, IHI, Z, LDZ, WORKL, NL, INFO )
|
|
IF( WANTT .OR. INFO.NE.0 )
|
|
$ CALL SLACPY( '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 SLASET( '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( REAL( MAX( 1, N ) ), WORK( 1 ) )
|
|
END IF
|
|
*
|
|
* ==== End of SHSEQR ====
|
|
*
|
|
END
|