*> \brief \b ZHSEQR * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZHSEQR + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, * WORK, LWORK, INFO ) * * .. Scalar Arguments .. * INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N * CHARACTER COMPZ, JOB * .. * .. Array Arguments .. * COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZHSEQR computes the eigenvalues of a Hessenberg matrix H *> and, optionally, the matrices T and Z from the Schur decomposition *> H = Z T Z**H, where T is an upper triangular matrix (the *> Schur form), and Z is the unitary matrix of Schur vectors. *> *> Optionally Z may be postmultiplied into an input unitary *> 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 unitary matrix Q: A = Q*H*Q**H = (QZ)*T*(QZ)**H. *> \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 unitary 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 ZGEBAL, and then passed to ZGEHRD *> when the matrix output by ZGEBAL 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 COMPLEX*16 array, dimension (LDH,N) *> On entry, the upper Hessenberg matrix H. *> On exit, if INFO = 0 and JOB = 'S', H contains the upper *> triangular matrix T from the Schur decomposition (the *> Schur form). 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 ZHSEQR, 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] W *> \verbatim *> W is COMPLEX*16 array, dimension (N) *> The computed eigenvalues. If JOB = 'S', the eigenvalues are *> stored in the same order as on the diagonal of the Schur *> form returned in H, with W(i) = H(i,i). *> \endverbatim *> *> \param[in,out] Z *> \verbatim *> Z is COMPLEX*16 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 unitary 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 unitary matrix generated by ZUNGHR *> after the call to ZGEHRD 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 COMPLEX*16 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 ZHSEQR does a workspace query. *> In this case, ZHSEQR 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, ZHSEQR 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 a unitary matrix. The final *> value of H is upper Hessenberg and 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 unitary 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 unitary 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 complex16OTHERcomputational * *> \par Contributors: * ================== *> *> Karen Braman and Ralph Byers, Department of Mathematics, *> University of Kansas, USA * *> \par Further Details: * ===================== *> *> \verbatim *> *> Default values supplied by *> ILAENV(ISPEC,'ZHSEQR',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 ZLAHQR vs ZLAQR0 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 *> ZLAHQR 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 ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, 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 .. COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * ) * .. * * ===================================================================== * * .. Parameters .. * * ==== Matrices of order NTINY or smaller must be processed by * . ZLAHQR 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 ZLAHQR 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 ) COMPLEX*16 ZERO, ONE PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ), \$ ONE = ( 1.0d0, 0.0d0 ) ) DOUBLE PRECISION RZERO PARAMETER ( RZERO = 0.0d0 ) * .. * .. Local Arrays .. COMPLEX*16 HL( NL, NL ), WORKL( NL ) * .. * .. Local Scalars .. INTEGER KBOT, NMIN LOGICAL INITZ, LQUERY, WANTT, WANTZ * .. * .. External Functions .. INTEGER ILAENV LOGICAL LSAME EXTERNAL ILAENV, LSAME * .. * .. External Subroutines .. EXTERNAL XERBLA, ZCOPY, ZLACPY, ZLAHQR, ZLAQR0, ZLASET * .. * .. Intrinsic Functions .. INTRINSIC DBLE, DCMPLX, 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 ) = DCMPLX( DBLE( MAX( 1, N ) ), RZERO ) 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 = -10 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN INFO = -12 END IF * IF( INFO.NE.0 ) THEN * * ==== Quick return in case of invalid argument. ==== * CALL XERBLA( 'ZHSEQR', -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 ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI, Z, \$ LDZ, WORK, LWORK, INFO ) * ==== Ensure reported workspace size is backward-compatible with * . previous LAPACK versions. ==== WORK( 1 ) = DCMPLX( MAX( DBLE( WORK( 1 ) ), DBLE( MAX( 1, \$ N ) ) ), RZERO ) RETURN * ELSE * * ==== copy eigenvalues isolated by ZGEBAL ==== * IF( ILO.GT.1 ) \$ CALL ZCOPY( ILO-1, H, LDH+1, W, 1 ) IF( IHI.LT.N ) \$ CALL ZCOPY( N-IHI, H( IHI+1, IHI+1 ), LDH+1, W( IHI+1 ), 1 ) * * ==== Initialize Z, if requested ==== * IF( INITZ ) \$ CALL ZLASET( 'A', N, N, ZERO, ONE, Z, LDZ ) * * ==== Quick return if possible ==== * IF( ILO.EQ.IHI ) THEN W( ILO ) = H( ILO, ILO ) RETURN END IF * * ==== ZLAHQR/ZLAQR0 crossover point ==== * NMIN = ILAENV( 12, 'ZHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N, \$ ILO, IHI, LWORK ) NMIN = MAX( NTINY, NMIN ) * * ==== ZLAQR0 for big matrices; ZLAHQR for small ones ==== * IF( N.GT.NMIN ) THEN CALL ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI, \$ Z, LDZ, WORK, LWORK, INFO ) ELSE * * ==== Small matrix ==== * CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI, \$ Z, LDZ, INFO ) * IF( INFO.GT.0 ) THEN * * ==== A rare ZLAHQR failure! ZLAQR0 sometimes succeeds * . when ZLAHQR fails. ==== * KBOT = INFO * IF( N.GE.NL ) THEN * * ==== Larger matrices have enough subdiagonal scratch * . space to call ZLAQR0 directly. ==== * CALL ZLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, W, \$ ILO, IHI, Z, LDZ, WORK, LWORK, INFO ) * ELSE * * ==== Tiny matrices don't have enough subdiagonal * . scratch space to benefit from ZLAQR0. Hence, * . tiny matrices must be copied into a larger * . array before calling ZLAQR0. ==== * CALL ZLACPY( 'A', N, N, H, LDH, HL, NL ) HL( N+1, N ) = ZERO CALL ZLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ), \$ NL ) CALL ZLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, W, \$ ILO, IHI, Z, LDZ, WORKL, NL, INFO ) IF( WANTT .OR. INFO.NE.0 ) \$ CALL ZLACPY( '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 ZLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH ) * * ==== Ensure reported workspace size is backward-compatible with * . previous LAPACK versions. ==== * WORK( 1 ) = DCMPLX( MAX( DBLE( MAX( 1, N ) ), \$ DBLE( WORK( 1 ) ) ), RZERO ) END IF * * ==== End of ZHSEQR ==== * END