--- rpl/lapack/lapack/dgeesx.f 2010/01/26 15:22:46 1.1.1.1
+++ rpl/lapack/lapack/dgeesx.f 2023/08/07 08:38:48 1.20
@@ -1,11 +1,287 @@
+*> \brief DGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download DGEESX + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE DGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,
+* WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK,
+* IWORK, LIWORK, BWORK, INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER JOBVS, SENSE, SORT
+* INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
+* DOUBLE PRECISION RCONDE, RCONDV
+* ..
+* .. Array Arguments ..
+* LOGICAL BWORK( * )
+* INTEGER IWORK( * )
+* DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
+* $ WR( * )
+* ..
+* .. Function Arguments ..
+* LOGICAL SELECT
+* EXTERNAL SELECT
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> DGEESX computes for an N-by-N real nonsymmetric matrix A, the
+*> eigenvalues, the real Schur form T, and, optionally, the matrix of
+*> Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T).
+*>
+*> Optionally, it also orders the eigenvalues on the diagonal of the
+*> real Schur form so that selected eigenvalues are at the top left;
+*> computes a reciprocal condition number for the average of the
+*> selected eigenvalues (RCONDE); and computes a reciprocal condition
+*> number for the right invariant subspace corresponding to the
+*> selected eigenvalues (RCONDV). The leading columns of Z form an
+*> orthonormal basis for this invariant subspace.
+*>
+*> For further explanation of the reciprocal condition numbers RCONDE
+*> and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
+*> these quantities are called s and sep respectively).
+*>
+*> A real matrix is in real Schur form if it is upper quasi-triangular
+*> with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in
+*> the form
+*> [ a b ]
+*> [ c a ]
+*>
+*> where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] JOBVS
+*> \verbatim
+*> JOBVS is CHARACTER*1
+*> = 'N': Schur vectors are not computed;
+*> = 'V': Schur vectors are computed.
+*> \endverbatim
+*>
+*> \param[in] SORT
+*> \verbatim
+*> SORT is CHARACTER*1
+*> Specifies whether or not to order the eigenvalues on the
+*> diagonal of the Schur form.
+*> = 'N': Eigenvalues are not ordered;
+*> = 'S': Eigenvalues are ordered (see SELECT).
+*> \endverbatim
+*>
+*> \param[in] SELECT
+*> \verbatim
+*> SELECT is a LOGICAL FUNCTION of two DOUBLE PRECISION arguments
+*> SELECT must be declared EXTERNAL in the calling subroutine.
+*> If SORT = 'S', SELECT is used to select eigenvalues to sort
+*> to the top left of the Schur form.
+*> If SORT = 'N', SELECT is not referenced.
+*> An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
+*> SELECT(WR(j),WI(j)) is true; i.e., if either one of a
+*> complex conjugate pair of eigenvalues is selected, then both
+*> are. Note that a selected complex eigenvalue may no longer
+*> satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
+*> ordering may change the value of complex eigenvalues
+*> (especially if the eigenvalue is ill-conditioned); in this
+*> case INFO may be set to N+3 (see INFO below).
+*> \endverbatim
+*>
+*> \param[in] SENSE
+*> \verbatim
+*> SENSE is CHARACTER*1
+*> Determines which reciprocal condition numbers are computed.
+*> = 'N': None are computed;
+*> = 'E': Computed for average of selected eigenvalues only;
+*> = 'V': Computed for selected right invariant subspace only;
+*> = 'B': Computed for both.
+*> If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The order of the matrix A. N >= 0.
+*> \endverbatim
+*>
+*> \param[in,out] A
+*> \verbatim
+*> A is DOUBLE PRECISION array, dimension (LDA, N)
+*> On entry, the N-by-N matrix A.
+*> On exit, A is overwritten by its real Schur form T.
+*> \endverbatim
+*>
+*> \param[in] LDA
+*> \verbatim
+*> LDA is INTEGER
+*> The leading dimension of the array A. LDA >= max(1,N).
+*> \endverbatim
+*>
+*> \param[out] SDIM
+*> \verbatim
+*> SDIM is INTEGER
+*> If SORT = 'N', SDIM = 0.
+*> If SORT = 'S', SDIM = number of eigenvalues (after sorting)
+*> for which SELECT is true. (Complex conjugate
+*> pairs for which SELECT is true for either
+*> eigenvalue count as 2.)
+*> \endverbatim
+*>
+*> \param[out] WR
+*> \verbatim
+*> WR is DOUBLE PRECISION array, dimension (N)
+*> \endverbatim
+*>
+*> \param[out] WI
+*> \verbatim
+*> WI is DOUBLE PRECISION array, dimension (N)
+*> WR and WI contain the real and imaginary parts, respectively,
+*> of the computed eigenvalues, in the same order that they
+*> appear on the diagonal of the output Schur form T. Complex
+*> conjugate pairs of eigenvalues appear consecutively with the
+*> eigenvalue having the positive imaginary part first.
+*> \endverbatim
+*>
+*> \param[out] VS
+*> \verbatim
+*> VS is DOUBLE PRECISION array, dimension (LDVS,N)
+*> If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
+*> vectors.
+*> If JOBVS = 'N', VS is not referenced.
+*> \endverbatim
+*>
+*> \param[in] LDVS
+*> \verbatim
+*> LDVS is INTEGER
+*> The leading dimension of the array VS. LDVS >= 1, and if
+*> JOBVS = 'V', LDVS >= N.
+*> \endverbatim
+*>
+*> \param[out] RCONDE
+*> \verbatim
+*> RCONDE is DOUBLE PRECISION
+*> If SENSE = 'E' or 'B', RCONDE contains the reciprocal
+*> condition number for the average of the selected eigenvalues.
+*> Not referenced if SENSE = 'N' or 'V'.
+*> \endverbatim
+*>
+*> \param[out] RCONDV
+*> \verbatim
+*> RCONDV is DOUBLE PRECISION
+*> If SENSE = 'V' or 'B', RCONDV contains the reciprocal
+*> condition number for the selected right invariant subspace.
+*> Not referenced if SENSE = 'N' or 'E'.
+*> \endverbatim
+*>
+*> \param[out] WORK
+*> \verbatim
+*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
+*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
+*> \endverbatim
+*>
+*> \param[in] LWORK
+*> \verbatim
+*> LWORK is INTEGER
+*> The dimension of the array WORK. LWORK >= max(1,3*N).
+*> Also, if SENSE = 'E' or 'V' or 'B',
+*> LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of
+*> selected eigenvalues computed by this routine. Note that
+*> N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only
+*> returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or
+*> 'B' this may not be large enough.
+*> For good performance, LWORK must generally be larger.
+*>
+*> If LWORK = -1, then a workspace query is assumed; the routine
+*> only calculates upper bounds on the optimal sizes of the
+*> arrays WORK and IWORK, returns these values as the first
+*> entries of the WORK and IWORK arrays, and no error messages
+*> related to LWORK or LIWORK are issued by XERBLA.
+*> \endverbatim
+*>
+*> \param[out] IWORK
+*> \verbatim
+*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
+*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
+*> \endverbatim
+*>
+*> \param[in] LIWORK
+*> \verbatim
+*> LIWORK is INTEGER
+*> The dimension of the array IWORK.
+*> LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
+*> Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is
+*> only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
+*> may not be large enough.
+*>
+*> If LIWORK = -1, then a workspace query is assumed; the
+*> routine only calculates upper bounds on the optimal sizes of
+*> the arrays WORK and IWORK, returns these values as the first
+*> entries of the WORK and IWORK arrays, and no error messages
+*> related to LWORK or LIWORK are issued by XERBLA.
+*> \endverbatim
+*>
+*> \param[out] BWORK
+*> \verbatim
+*> BWORK is LOGICAL array, dimension (N)
+*> Not referenced if SORT = 'N'.
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> INFO is INTEGER
+*> = 0: successful exit
+*> < 0: if INFO = -i, the i-th argument had an illegal value.
+*> > 0: if INFO = i, and i is
+*> <= N: the QR algorithm failed to compute all the
+*> eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
+*> contain those eigenvalues which have converged; if
+*> JOBVS = 'V', VS contains the transformation which
+*> reduces A to its partially converged Schur form.
+*> = N+1: the eigenvalues could not be reordered because some
+*> eigenvalues were too close to separate (the problem
+*> is very ill-conditioned);
+*> = N+2: after reordering, roundoff changed values of some
+*> complex eigenvalues so that leading eigenvalues in
+*> the Schur form no longer satisfy SELECT=.TRUE. This
+*> could also be caused by underflow due to scaling.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \ingroup doubleGEeigen
+*
+* =====================================================================
SUBROUTINE DGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,
$ WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK,
$ IWORK, LIWORK, BWORK, INFO )
*
-* -- LAPACK driver routine (version 3.2) --
+* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* November 2006
*
* .. Scalar Arguments ..
CHARACTER JOBVS, SENSE, SORT
@@ -23,167 +299,6 @@
EXTERNAL SELECT
* ..
*
-* Purpose
-* =======
-*
-* DGEESX computes for an N-by-N real nonsymmetric matrix A, the
-* eigenvalues, the real Schur form T, and, optionally, the matrix of
-* Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T).
-*
-* Optionally, it also orders the eigenvalues on the diagonal of the
-* real Schur form so that selected eigenvalues are at the top left;
-* computes a reciprocal condition number for the average of the
-* selected eigenvalues (RCONDE); and computes a reciprocal condition
-* number for the right invariant subspace corresponding to the
-* selected eigenvalues (RCONDV). The leading columns of Z form an
-* orthonormal basis for this invariant subspace.
-*
-* For further explanation of the reciprocal condition numbers RCONDE
-* and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
-* these quantities are called s and sep respectively).
-*
-* A real matrix is in real Schur form if it is upper quasi-triangular
-* with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in
-* the form
-* [ a b ]
-* [ c a ]
-*
-* where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
-*
-* Arguments
-* =========
-*
-* JOBVS (input) CHARACTER*1
-* = 'N': Schur vectors are not computed;
-* = 'V': Schur vectors are computed.
-*
-* SORT (input) CHARACTER*1
-* Specifies whether or not to order the eigenvalues on the
-* diagonal of the Schur form.
-* = 'N': Eigenvalues are not ordered;
-* = 'S': Eigenvalues are ordered (see SELECT).
-*
-* SELECT (external procedure) LOGICAL FUNCTION of two DOUBLE PRECISION arguments
-* SELECT must be declared EXTERNAL in the calling subroutine.
-* If SORT = 'S', SELECT is used to select eigenvalues to sort
-* to the top left of the Schur form.
-* If SORT = 'N', SELECT is not referenced.
-* An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
-* SELECT(WR(j),WI(j)) is true; i.e., if either one of a
-* complex conjugate pair of eigenvalues is selected, then both
-* are. Note that a selected complex eigenvalue may no longer
-* satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
-* ordering may change the value of complex eigenvalues
-* (especially if the eigenvalue is ill-conditioned); in this
-* case INFO may be set to N+3 (see INFO below).
-*
-* SENSE (input) CHARACTER*1
-* Determines which reciprocal condition numbers are computed.
-* = 'N': None are computed;
-* = 'E': Computed for average of selected eigenvalues only;
-* = 'V': Computed for selected right invariant subspace only;
-* = 'B': Computed for both.
-* If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
-*
-* N (input) INTEGER
-* The order of the matrix A. N >= 0.
-*
-* A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
-* On entry, the N-by-N matrix A.
-* On exit, A is overwritten by its real Schur form T.
-*
-* LDA (input) INTEGER
-* The leading dimension of the array A. LDA >= max(1,N).
-*
-* SDIM (output) INTEGER
-* If SORT = 'N', SDIM = 0.
-* If SORT = 'S', SDIM = number of eigenvalues (after sorting)
-* for which SELECT is true. (Complex conjugate
-* pairs for which SELECT is true for either
-* eigenvalue count as 2.)
-*
-* WR (output) DOUBLE PRECISION array, dimension (N)
-* WI (output) DOUBLE PRECISION array, dimension (N)
-* WR and WI contain the real and imaginary parts, respectively,
-* of the computed eigenvalues, in the same order that they
-* appear on the diagonal of the output Schur form T. Complex
-* conjugate pairs of eigenvalues appear consecutively with the
-* eigenvalue having the positive imaginary part first.
-*
-* VS (output) DOUBLE PRECISION array, dimension (LDVS,N)
-* If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
-* vectors.
-* If JOBVS = 'N', VS is not referenced.
-*
-* LDVS (input) INTEGER
-* The leading dimension of the array VS. LDVS >= 1, and if
-* JOBVS = 'V', LDVS >= N.
-*
-* RCONDE (output) DOUBLE PRECISION
-* If SENSE = 'E' or 'B', RCONDE contains the reciprocal
-* condition number for the average of the selected eigenvalues.
-* Not referenced if SENSE = 'N' or 'V'.
-*
-* RCONDV (output) DOUBLE PRECISION
-* If SENSE = 'V' or 'B', RCONDV contains the reciprocal
-* condition number for the selected right invariant subspace.
-* Not referenced if SENSE = 'N' or 'E'.
-*
-* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
-* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
-*
-* LWORK (input) INTEGER
-* The dimension of the array WORK. LWORK >= max(1,3*N).
-* Also, if SENSE = 'E' or 'V' or 'B',
-* LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of
-* selected eigenvalues computed by this routine. Note that
-* N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only
-* returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or
-* 'B' this may not be large enough.
-* For good performance, LWORK must generally be larger.
-*
-* If LWORK = -1, then a workspace query is assumed; the routine
-* only calculates upper bounds on the optimal sizes of the
-* arrays WORK and IWORK, returns these values as the first
-* entries of the WORK and IWORK arrays, and no error messages
-* related to LWORK or LIWORK are issued by XERBLA.
-*
-* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
-* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
-*
-* LIWORK (input) INTEGER
-* The dimension of the array IWORK.
-* LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
-* Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is
-* only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
-* may not be large enough.
-*
-* If LIWORK = -1, then a workspace query is assumed; the
-* routine only calculates upper bounds on the optimal sizes of
-* the arrays WORK and IWORK, returns these values as the first
-* entries of the WORK and IWORK arrays, and no error messages
-* related to LWORK or LIWORK are issued by XERBLA.
-*
-* BWORK (workspace) LOGICAL array, dimension (N)
-* Not referenced if SORT = 'N'.
-*
-* INFO (output) INTEGER
-* = 0: successful exit
-* < 0: if INFO = -i, the i-th argument had an illegal value.
-* > 0: if INFO = i, and i is
-* <= N: the QR algorithm failed to compute all the
-* eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
-* contain those eigenvalues which have converged; if
-* JOBVS = 'V', VS contains the transformation which
-* reduces A to its partially converged Schur form.
-* = N+1: the eigenvalues could not be reordered because some
-* eigenvalues were too close to separate (the problem
-* is very ill-conditioned);
-* = N+2: after reordering, roundoff changed values of some
-* complex eigenvalues so that leading eigenvalues in
-* the Schur form no longer satisfy SELECT=.TRUE. This
-* could also be caused by underflow due to scaling.
-*
* =====================================================================
*
* .. Parameters ..
@@ -226,6 +341,7 @@
WANTSV = LSAME( SENSE, 'V' )
WANTSB = LSAME( SENSE, 'B' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
+*
IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN
INFO = -1
ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
@@ -266,7 +382,7 @@
*
CALL DHSEQR( 'S', JOBVS, N, 1, N, A, LDA, WR, WI, VS, LDVS,
$ WORK, -1, IEVAL )
- HSWORK = WORK( 1 )
+ HSWORK = INT( WORK( 1 ) )
*
IF( .NOT.WANTVS ) THEN
MAXWRK = MAX( MAXWRK, N + HSWORK )
@@ -294,6 +410,8 @@
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEESX', -INFO )
RETURN
+ ELSE IF( LQUERY ) THEN
+ RETURN
END IF
*
* Quick return if possible
@@ -462,7 +580,9 @@
IF( N.GT.I+1 )
$ CALL DSWAP( N-I-1, A( I, I+2 ), LDA,
$ A( I+1, I+2 ), LDA )
- CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ), 1 )
+ IF( WANTVS ) THEN
+ CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ), 1 )
+ END IF
A( I, I+1 ) = A( I+1, I )
A( I+1, I ) = ZERO
END IF