#include "ESMF_LapackBlas.inc" !> \brief \b DHSEQR ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DHSEQR + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhseqr.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhseqr.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhseqr.f"> !> [TXT]</a> !> \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 December 2016 ! !> \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 3.7.0) -- ! -- LAPACK is a software package provided by Univ. of Tennessee, -- ! -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- ! December 2016 ! ! .. 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