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dhseqr
C DHSEQR    SOURCE    BP208322  20/09/18    21:15:55     10718          *> \brief \b DHSEQR**  =========== DOCUMENTATION ===========** Online html documentation available at*            http://www.netlib.org/lapack/explore-html/**> \htmlonly*> Download DHSEQR + dependencies*> &lt;a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhseqr.f">*> [TGZ]&lt;/a>*> &lt;a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhseqr.f">*> [ZIP]&lt;/a>*> &lt;a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhseqr.f">*> [TXT]&lt;/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 ..*       REAL*8   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 REAL*8 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 REAL*8 array, dimension (N)*> \endverbatim*>*> \param[out] WI*> \verbatim*> WI is REAL*8 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 REAL*8 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 REAL*8 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       IMPLICIT INTEGER(I-N)      IMPLICIT REAL*8(A-H,O-Z)**     .. Scalar Arguments ..      INTEGER            IHI, ILO, INFO, LDH, LDZ, LWORK, N      CHARACTER          COMPZ, JOB*     ..*     .. Array Arguments ..      REAL*8   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 ) REAL*8 ZERO, ONE PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )* ..* .. Local Arrays .. REAL*8 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   

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