version 1.1, 2010/01/26 15:22:45
|
version 1.19, 2023/08/07 08:38:56
|
Line 1
|
Line 1
|
|
*> \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 |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr4.f"> |
|
*> [TGZ]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr4.f"> |
|
*> [ZIP]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr4.f"> |
|
*> [TXT]</a> |
|
*> \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 >= 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 > 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 > 0, then 1 <= ILO <= IHI <= 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) < 0. If INFO = 0 and WANTT is |
|
*> .FALSE., then the contents of H are unspecified on exit. |
|
*> (The output value of H when INFO > 0 is given under the |
|
*> description of INFO below.) |
|
*> |
|
*> This subroutine may explicitly set 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 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) > 0 and WI(i+1) < 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 <= ILOZ <= ILO; IHI <= IHIZ <= 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 > 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 >= MAX(1,IHIZ). Otherwise, LDZ >= 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 >= 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 |
|
*> > 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 > 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 > 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 > 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 > 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. |
|
* |
|
*> \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, |
SUBROUTINE DLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, |
$ ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO ) |
$ ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO ) |
* |
* |
* -- LAPACK auxiliary routine (version 3.2) -- |
* -- LAPACK auxiliary routine -- |
* Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd.. |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* November 2006 |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N |
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N |
Line 14
|
Line 274
|
$ Z( LDZ, * ) |
$ Z( LDZ, * ) |
* .. |
* .. |
* |
* |
* This subroutine 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. |
|
* |
|
* Purpose |
|
* ======= |
|
* |
|
* 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. |
|
* |
|
* Arguments |
|
* ========= |
|
* |
|
* WANTT (input) LOGICAL |
|
* = .TRUE. : the full Schur form T is required; |
|
* = .FALSE.: only eigenvalues are required. |
|
* |
|
* WANTZ (input) LOGICAL |
|
* = .TRUE. : the matrix of Schur vectors Z is required; |
|
* = .FALSE.: Schur vectors are not required. |
|
* |
|
* N (input) INTEGER |
|
* The order of the matrix H. N .GE. 0. |
|
* |
|
* ILO (input) INTEGER |
|
* IHI (input) 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. |
|
* |
|
* H (input/output) 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. |
|
* |
|
* LDH (input) INTEGER |
|
* The leading dimension of the array H. LDH .GE. max(1,N). |
|
* |
|
* WR (output) DOUBLE PRECISION array, dimension (IHI) |
|
* WI (output) 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). |
|
* |
|
* ILOZ (input) INTEGER |
|
* IHIZ (input) 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. |
|
* |
|
* Z (input/output) 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.) |
|
* |
|
* LDZ (input) INTEGER |
|
* The leading dimension of the array Z. if WANTZ is .TRUE. |
|
* then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1. |
|
* |
|
* WORK (workspace/output) DOUBLE PRECISION array, dimension LWORK |
|
* On exit, if LWORK = -1, WORK(1) returns an estimate of |
|
* the optimal value for LWORK. |
|
* |
|
* LWORK (input) 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. |
|
* |
|
* |
|
* INFO (output) 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 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. |
|
* |
|
* ================================================================ |
|
* Based on contributions by |
|
* Karen Braman and Ralph Byers, Department of Mathematics, |
|
* University of Kansas, USA |
|
* |
|
* ================================================================ |
|
* 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. |
|
* |
|
* 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. |
|
* |
|
* ================================================================ |
|
* .. Parameters .. |
* .. Parameters .. |
* |
* |
* ==== Matrices of order NTINY or smaller must be processed by |
* ==== Matrices of order NTINY or smaller must be processed by |
* . DLAHQR because of insufficient subdiagonal scratch space. |
* . DLAHQR because of insufficient subdiagonal scratch space. |
* . (This is a hard limit.) ==== |
* . (This is a hard limit.) ==== |
INTEGER NTINY |
INTEGER NTINY |
PARAMETER ( NTINY = 11 ) |
PARAMETER ( NTINY = 15 ) |
* |
* |
* ==== Exceptional deflation windows: try to cure rare |
* ==== Exceptional deflation windows: try to cure rare |
* . slow convergence by varying the size of the |
* . slow convergence by varying the size of the |
Line 266
|
Line 365
|
END IF |
END IF |
* |
* |
* ==== NWR = recommended deflation window size. At this |
* ==== NWR = recommended deflation window size. At this |
* . point, N .GT. NTINY = 11, so there is enough |
* . point, N .GT. NTINY = 15, so there is enough |
* . subdiagonal workspace for NWR.GE.2 as required. |
* . subdiagonal workspace for NWR.GE.2 as required. |
* . (In fact, there is enough subdiagonal space for |
* . (In fact, there is enough subdiagonal space for |
* . NWR.GE.3.) ==== |
* . NWR.GE.4.) ==== |
* |
* |
NWR = ILAENV( 13, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK ) |
NWR = ILAENV( 13, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK ) |
NWR = MAX( 2, NWR ) |
NWR = MAX( 2, NWR ) |
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR ) |
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR ) |
* |
* |
* ==== NSR = recommended number of simultaneous shifts. |
* ==== NSR = recommended number of simultaneous shifts. |
* . At this point N .GT. NTINY = 11, so there is at |
* . At this point N .GT. NTINY = 15, so there is at |
* . enough subdiagonal workspace for NSR to be even |
* . enough subdiagonal workspace for NSR to be even |
* . and greater than or equal to two as required. ==== |
* . and greater than or equal to two as required. ==== |
* |
* |
NSR = ILAENV( 15, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK ) |
NSR = ILAENV( 15, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK ) |
NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO ) |
NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO ) |
NSR = MAX( 2, NSR-MOD( NSR, 2 ) ) |
NSR = MAX( 2, NSR-MOD( NSR, 2 ) ) |
* |
* |
* ==== Estimate optimal workspace ==== |
* ==== Estimate optimal workspace ==== |
Line 329
|
Line 428
|
* ==== NSMAX = the Largest number of simultaneous shifts |
* ==== NSMAX = the Largest number of simultaneous shifts |
* . for which there is sufficient workspace. ==== |
* . for which there is sufficient workspace. ==== |
* |
* |
NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 ) |
NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 ) |
NSMAX = NSMAX - MOD( NSMAX, 2 ) |
NSMAX = NSMAX - MOD( NSMAX, 2 ) |
* |
* |
* ==== NDFL: an iteration count restarted at deflation. ==== |
* ==== NDFL: an iteration count restarted at deflation. ==== |
Line 480
|
Line 579
|
* |
* |
* ==== Got NS/2 or fewer shifts? Use DLAHQR |
* ==== Got NS/2 or fewer shifts? Use DLAHQR |
* . on a trailing principal submatrix to |
* . on a trailing principal submatrix to |
* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9, |
* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6, |
* . there is enough space below the subdiagonal |
* . there is enough space below the subdiagonal |
* . to fit an NS-by-NS scratch array.) ==== |
* . to fit an NS-by-NS scratch array.) ==== |
* |
* |
Line 575
|
Line 674
|
END IF |
END IF |
END IF |
END IF |
* |
* |
* ==== Use up to NS of the the smallest magnatiude |
* ==== Use up to NS of the the smallest magnitude |
* . shifts. If there aren't NS shifts available, |
* . shifts. If there aren't NS shifts available, |
* . then use them all, possibly dropping one to |
* . then use them all, possibly dropping one to |
* . make the number of shifts even. ==== |
* . make the number of shifts even. ==== |
Line 595
|
Line 694
|
* . (NVE-by-KDU) vertical work WV arrow along |
* . (NVE-by-KDU) vertical work WV arrow along |
* . the left-hand-edge. ==== |
* . the left-hand-edge. ==== |
* |
* |
KDU = 3*NS - 3 |
KDU = 2*NS |
KU = N - KDU + 1 |
KU = N - KDU + 1 |
KWH = KDU + 1 |
KWH = KDU + 1 |
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1 |
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1 |