--- rpl/lapack/lapack/zggesx.f 2010/08/07 13:22:32 1.5
+++ rpl/lapack/lapack/zggesx.f 2023/08/07 08:39:20 1.19
@@ -1,12 +1,336 @@
+*> \brief ZGGESX 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 ZGGESX + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE ZGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA,
+* B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR,
+* LDVSR, RCONDE, RCONDV, WORK, LWORK, RWORK,
+* IWORK, LIWORK, BWORK, INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER JOBVSL, JOBVSR, SENSE, SORT
+* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
+* $ SDIM
+* ..
+* .. Array Arguments ..
+* LOGICAL BWORK( * )
+* INTEGER IWORK( * )
+* DOUBLE PRECISION RCONDE( 2 ), RCONDV( 2 ), RWORK( * )
+* COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
+* $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
+* $ WORK( * )
+* ..
+* .. Function Arguments ..
+* LOGICAL SELCTG
+* EXTERNAL SELCTG
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> ZGGESX computes for a pair of N-by-N complex nonsymmetric matrices
+*> (A,B), the generalized eigenvalues, the complex Schur form (S,T),
+*> and, optionally, the left and/or right matrices of Schur vectors (VSL
+*> and VSR). This gives the generalized Schur factorization
+*>
+*> (A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H )
+*>
+*> where (VSR)**H is the conjugate-transpose of VSR.
+*>
+*> Optionally, it also orders the eigenvalues so that a selected cluster
+*> of eigenvalues appears in the leading diagonal blocks of the upper
+*> triangular matrix S and the upper triangular matrix T; computes
+*> a reciprocal condition number for the average of the selected
+*> eigenvalues (RCONDE); and computes a reciprocal condition number for
+*> the right and left deflating subspaces corresponding to the selected
+*> eigenvalues (RCONDV). The leading columns of VSL and VSR then form
+*> an orthonormal basis for the corresponding left and right eigenspaces
+*> (deflating subspaces).
+*>
+*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
+*> or a ratio alpha/beta = w, such that A - w*B is singular. It is
+*> usually represented as the pair (alpha,beta), as there is a
+*> reasonable interpretation for beta=0 or for both being zero.
+*>
+*> A pair of matrices (S,T) is in generalized complex Schur form if T is
+*> upper triangular with non-negative diagonal and S is upper
+*> triangular.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] JOBVSL
+*> \verbatim
+*> JOBVSL is CHARACTER*1
+*> = 'N': do not compute the left Schur vectors;
+*> = 'V': compute the left Schur vectors.
+*> \endverbatim
+*>
+*> \param[in] JOBVSR
+*> \verbatim
+*> JOBVSR is CHARACTER*1
+*> = 'N': do not compute the right Schur vectors;
+*> = 'V': compute the right Schur vectors.
+*> \endverbatim
+*>
+*> \param[in] SORT
+*> \verbatim
+*> SORT is CHARACTER*1
+*> Specifies whether or not to order the eigenvalues on the
+*> diagonal of the generalized Schur form.
+*> = 'N': Eigenvalues are not ordered;
+*> = 'S': Eigenvalues are ordered (see SELCTG).
+*> \endverbatim
+*>
+*> \param[in] SELCTG
+*> \verbatim
+*> SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 arguments
+*> SELCTG must be declared EXTERNAL in the calling subroutine.
+*> If SORT = 'N', SELCTG is not referenced.
+*> If SORT = 'S', SELCTG is used to select eigenvalues to sort
+*> to the top left of the Schur form.
+*> Note that a selected complex eigenvalue may no longer satisfy
+*> SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
+*> ordering may change the value of complex eigenvalues
+*> (especially if the eigenvalue is ill-conditioned), in this
+*> case INFO is 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 deflating subspaces 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 matrices A, B, VSL, and VSR. N >= 0.
+*> \endverbatim
+*>
+*> \param[in,out] A
+*> \verbatim
+*> A is COMPLEX*16 array, dimension (LDA, N)
+*> On entry, the first of the pair of matrices.
+*> On exit, A has been overwritten by its generalized Schur
+*> form S.
+*> \endverbatim
+*>
+*> \param[in] LDA
+*> \verbatim
+*> LDA is INTEGER
+*> The leading dimension of A. LDA >= max(1,N).
+*> \endverbatim
+*>
+*> \param[in,out] B
+*> \verbatim
+*> B is COMPLEX*16 array, dimension (LDB, N)
+*> On entry, the second of the pair of matrices.
+*> On exit, B has been overwritten by its generalized Schur
+*> form T.
+*> \endverbatim
+*>
+*> \param[in] LDB
+*> \verbatim
+*> LDB is INTEGER
+*> The leading dimension of B. LDB >= 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 SELCTG is true.
+*> \endverbatim
+*>
+*> \param[out] ALPHA
+*> \verbatim
+*> ALPHA is COMPLEX*16 array, dimension (N)
+*> \endverbatim
+*>
+*> \param[out] BETA
+*> \verbatim
+*> BETA is COMPLEX*16 array, dimension (N)
+*> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
+*> generalized eigenvalues. ALPHA(j) and BETA(j),j=1,...,N are
+*> the diagonals of the complex Schur form (S,T). BETA(j) will
+*> be non-negative real.
+*>
+*> Note: the quotients ALPHA(j)/BETA(j) may easily over- or
+*> underflow, and BETA(j) may even be zero. Thus, the user
+*> should avoid naively computing the ratio alpha/beta.
+*> However, ALPHA will be always less than and usually
+*> comparable with norm(A) in magnitude, and BETA always less
+*> than and usually comparable with norm(B).
+*> \endverbatim
+*>
+*> \param[out] VSL
+*> \verbatim
+*> VSL is COMPLEX*16 array, dimension (LDVSL,N)
+*> If JOBVSL = 'V', VSL will contain the left Schur vectors.
+*> Not referenced if JOBVSL = 'N'.
+*> \endverbatim
+*>
+*> \param[in] LDVSL
+*> \verbatim
+*> LDVSL is INTEGER
+*> The leading dimension of the matrix VSL. LDVSL >=1, and
+*> if JOBVSL = 'V', LDVSL >= N.
+*> \endverbatim
+*>
+*> \param[out] VSR
+*> \verbatim
+*> VSR is COMPLEX*16 array, dimension (LDVSR,N)
+*> If JOBVSR = 'V', VSR will contain the right Schur vectors.
+*> Not referenced if JOBVSR = 'N'.
+*> \endverbatim
+*>
+*> \param[in] LDVSR
+*> \verbatim
+*> LDVSR is INTEGER
+*> The leading dimension of the matrix VSR. LDVSR >= 1, and
+*> if JOBVSR = 'V', LDVSR >= N.
+*> \endverbatim
+*>
+*> \param[out] RCONDE
+*> \verbatim
+*> RCONDE is DOUBLE PRECISION array, dimension ( 2 )
+*> If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
+*> reciprocal condition numbers for the average of the selected
+*> eigenvalues.
+*> Not referenced if SENSE = 'N' or 'V'.
+*> \endverbatim
+*>
+*> \param[out] RCONDV
+*> \verbatim
+*> RCONDV is DOUBLE PRECISION array, dimension ( 2 )
+*> If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
+*> reciprocal condition number for the selected deflating
+*> subspaces.
+*> Not referenced if SENSE = 'N' or 'E'.
+*> \endverbatim
+*>
+*> \param[out] WORK
+*> \verbatim
+*> WORK is COMPLEX*16 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.
+*> If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
+*> LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)), else
+*> LWORK >= MAX(1,2*N). Note that 2*SDIM*(N-SDIM) <= N*N/2.
+*> Note also that an error is only returned if
+*> LWORK < MAX(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may
+*> not be large enough.
+*>
+*> If LWORK = -1, then a workspace query is assumed; the routine
+*> only calculates the bound on the optimal size of the WORK
+*> array and the minimum size of the IWORK array, returns these
+*> values as the first entries of the WORK and IWORK arrays, and
+*> no error message related to LWORK or LIWORK is issued by
+*> XERBLA.
+*> \endverbatim
+*>
+*> \param[out] RWORK
+*> \verbatim
+*> RWORK is DOUBLE PRECISION array, dimension ( 8*N )
+*> Real workspace.
+*> \endverbatim
+*>
+*> \param[out] IWORK
+*> \verbatim
+*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
+*> On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
+*> \endverbatim
+*>
+*> \param[in] LIWORK
+*> \verbatim
+*> LIWORK is INTEGER
+*> The dimension of the array IWORK.
+*> If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
+*> LIWORK >= N+2.
+*>
+*> If LIWORK = -1, then a workspace query is assumed; the
+*> routine only calculates the bound on the optimal size of the
+*> WORK array and the minimum size of the IWORK array, returns
+*> these values as the first entries of the WORK and IWORK
+*> arrays, and no error message related to LWORK or LIWORK is
+*> 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.
+*> = 1,...,N:
+*> The QZ iteration failed. (A,B) are not in Schur
+*> form, but ALPHA(j) and BETA(j) should be correct for
+*> j=INFO+1,...,N.
+*> > N: =N+1: other than QZ iteration failed in ZHGEQZ
+*> =N+2: after reordering, roundoff changed values of
+*> some complex eigenvalues so that leading
+*> eigenvalues in the Generalized Schur form no
+*> longer satisfy SELCTG=.TRUE. This could also
+*> be caused due to scaling.
+*> =N+3: reordering failed in ZTGSEN.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \ingroup complex16GEeigen
+*
+* =====================================================================
SUBROUTINE ZGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA,
$ B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR,
$ LDVSR, RCONDE, RCONDV, WORK, LWORK, RWORK,
$ 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 JOBVSL, JOBVSR, SENSE, SORT
@@ -26,194 +350,6 @@
EXTERNAL SELCTG
* ..
*
-* Purpose
-* =======
-*
-* ZGGESX computes for a pair of N-by-N complex nonsymmetric matrices
-* (A,B), the generalized eigenvalues, the complex Schur form (S,T),
-* and, optionally, the left and/or right matrices of Schur vectors (VSL
-* and VSR). This gives the generalized Schur factorization
-*
-* (A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H )
-*
-* where (VSR)**H is the conjugate-transpose of VSR.
-*
-* Optionally, it also orders the eigenvalues so that a selected cluster
-* of eigenvalues appears in the leading diagonal blocks of the upper
-* triangular matrix S and the upper triangular matrix T; computes
-* a reciprocal condition number for the average of the selected
-* eigenvalues (RCONDE); and computes a reciprocal condition number for
-* the right and left deflating subspaces corresponding to the selected
-* eigenvalues (RCONDV). The leading columns of VSL and VSR then form
-* an orthonormal basis for the corresponding left and right eigenspaces
-* (deflating subspaces).
-*
-* A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
-* or a ratio alpha/beta = w, such that A - w*B is singular. It is
-* usually represented as the pair (alpha,beta), as there is a
-* reasonable interpretation for beta=0 or for both being zero.
-*
-* A pair of matrices (S,T) is in generalized complex Schur form if T is
-* upper triangular with non-negative diagonal and S is upper
-* triangular.
-*
-* Arguments
-* =========
-*
-* JOBVSL (input) CHARACTER*1
-* = 'N': do not compute the left Schur vectors;
-* = 'V': compute the left Schur vectors.
-*
-* JOBVSR (input) CHARACTER*1
-* = 'N': do not compute the right Schur vectors;
-* = 'V': compute the right Schur vectors.
-*
-* SORT (input) CHARACTER*1
-* Specifies whether or not to order the eigenvalues on the
-* diagonal of the generalized Schur form.
-* = 'N': Eigenvalues are not ordered;
-* = 'S': Eigenvalues are ordered (see SELCTG).
-*
-* SELCTG (external procedure) LOGICAL FUNCTION of two COMPLEX*16 arguments
-* SELCTG must be declared EXTERNAL in the calling subroutine.
-* If SORT = 'N', SELCTG is not referenced.
-* If SORT = 'S', SELCTG is used to select eigenvalues to sort
-* to the top left of the Schur form.
-* Note that a selected complex eigenvalue may no longer satisfy
-* SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
-* ordering may change the value of complex eigenvalues
-* (especially if the eigenvalue is ill-conditioned), in this
-* case INFO is 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 deflating subspaces only;
-* = 'B' : Computed for both.
-* If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.
-*
-* N (input) INTEGER
-* The order of the matrices A, B, VSL, and VSR. N >= 0.
-*
-* A (input/output) COMPLEX*16 array, dimension (LDA, N)
-* On entry, the first of the pair of matrices.
-* On exit, A has been overwritten by its generalized Schur
-* form S.
-*
-* LDA (input) INTEGER
-* The leading dimension of A. LDA >= max(1,N).
-*
-* B (input/output) COMPLEX*16 array, dimension (LDB, N)
-* On entry, the second of the pair of matrices.
-* On exit, B has been overwritten by its generalized Schur
-* form T.
-*
-* LDB (input) INTEGER
-* The leading dimension of B. LDB >= max(1,N).
-*
-* SDIM (output) INTEGER
-* If SORT = 'N', SDIM = 0.
-* If SORT = 'S', SDIM = number of eigenvalues (after sorting)
-* for which SELCTG is true.
-*
-* ALPHA (output) COMPLEX*16 array, dimension (N)
-* BETA (output) COMPLEX*16 array, dimension (N)
-* On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
-* generalized eigenvalues. ALPHA(j) and BETA(j),j=1,...,N are
-* the diagonals of the complex Schur form (S,T). BETA(j) will
-* be non-negative real.
-*
-* Note: the quotients ALPHA(j)/BETA(j) may easily over- or
-* underflow, and BETA(j) may even be zero. Thus, the user
-* should avoid naively computing the ratio alpha/beta.
-* However, ALPHA will be always less than and usually
-* comparable with norm(A) in magnitude, and BETA always less
-* than and usually comparable with norm(B).
-*
-* VSL (output) COMPLEX*16 array, dimension (LDVSL,N)
-* If JOBVSL = 'V', VSL will contain the left Schur vectors.
-* Not referenced if JOBVSL = 'N'.
-*
-* LDVSL (input) INTEGER
-* The leading dimension of the matrix VSL. LDVSL >=1, and
-* if JOBVSL = 'V', LDVSL >= N.
-*
-* VSR (output) COMPLEX*16 array, dimension (LDVSR,N)
-* If JOBVSR = 'V', VSR will contain the right Schur vectors.
-* Not referenced if JOBVSR = 'N'.
-*
-* LDVSR (input) INTEGER
-* The leading dimension of the matrix VSR. LDVSR >= 1, and
-* if JOBVSR = 'V', LDVSR >= N.
-*
-* RCONDE (output) DOUBLE PRECISION array, dimension ( 2 )
-* If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
-* reciprocal condition numbers for the average of the selected
-* eigenvalues.
-* Not referenced if SENSE = 'N' or 'V'.
-*
-* RCONDV (output) DOUBLE PRECISION array, dimension ( 2 )
-* If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
-* reciprocal condition number for the selected deflating
-* subspaces.
-* Not referenced if SENSE = 'N' or 'E'.
-*
-* WORK (workspace/output) COMPLEX*16 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.
-* If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
-* LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)), else
-* LWORK >= MAX(1,2*N). Note that 2*SDIM*(N-SDIM) <= N*N/2.
-* Note also that an error is only returned if
-* LWORK < MAX(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may
-* not be large enough.
-*
-* If LWORK = -1, then a workspace query is assumed; the routine
-* only calculates the bound on the optimal size of the WORK
-* array and the minimum size of the IWORK array, returns these
-* values as the first entries of the WORK and IWORK arrays, and
-* no error message related to LWORK or LIWORK is issued by
-* XERBLA.
-*
-* RWORK (workspace) DOUBLE PRECISION array, dimension ( 8*N )
-* Real workspace.
-*
-* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
-* On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
-*
-* LIWORK (input) INTEGER
-* The dimension of the array IWORK.
-* If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
-* LIWORK >= N+2.
-*
-* If LIWORK = -1, then a workspace query is assumed; the
-* routine only calculates the bound on the optimal size of the
-* WORK array and the minimum size of the IWORK array, returns
-* these values as the first entries of the WORK and IWORK
-* arrays, and no error message related to LWORK or LIWORK is
-* 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.
-* = 1,...,N:
-* The QZ iteration failed. (A,B) are not in Schur
-* form, but ALPHA(j) and BETA(j) should be correct for
-* j=INFO+1,...,N.
-* > N: =N+1: other than QZ iteration failed in ZHGEQZ
-* =N+2: after reordering, roundoff changed values of
-* some complex eigenvalues so that leading
-* eigenvalues in the Generalized Schur form no
-* longer satisfy SELCTG=.TRUE. This could also
-* be caused due to scaling.
-* =N+3: reordering failed in ZTGSEN.
-*
* =====================================================================
*
* .. Parameters ..