--- rpl/lapack/lapack/dhseqr.f 2010/12/21 13:53:27 1.8
+++ rpl/lapack/lapack/dhseqr.f 2011/11/21 20:42:53 1.9
@@ -1,9 +1,325 @@
+*> \brief \b DHSEQR
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download DHSEQR + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \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 November 2011
+*
+*> \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 3.2.2) --
-* Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..
-* June 2010
+* -- LAPACK computational routine (version 3.4.0) --
+* -- LAPACK is a software package provided by Univ. of Tennessee, --
+* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
+* November 2011
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
@@ -13,221 +329,9 @@
DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
$ Z( LDZ, * )
* ..
-* Purpose
-* =======
*
-* 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.
-*
-* Arguments
-* =========
-*
-* JOB (input) CHARACTER*1
-* = 'E': compute eigenvalues only;
-* = 'S': compute eigenvalues and the Schur form T.
-*
-* COMPZ (input) 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.
-*
-* 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. 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 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.
-*
-* LDH (input) INTEGER
-* The leading dimension of the array H. LDH .GE. max(1,N).
-*
-* WR (output) DOUBLE PRECISION array, dimension (N)
-* WI (output) 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).
-*
-* Z (input/output) 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.)
-*
-* LDZ (input) INTEGER
-* The leading dimension of the array Z. if COMPZ = 'I' or
-* COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1.
-*
-* WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
-* On exit, if INFO = 0, 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 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.
-*
-*
-* INFO (output) 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.
-*
-* ================================================================
-* 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.
-*
-* ================================================================
-* 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 ..
*
* ==== Matrices of order NTINY or smaller must be processed by