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version 1.7, 2010/12/21 13:53:44 version 1.8, 2011/11/21 20:43:10
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   *> \brief <b> ZGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZGGES + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgges.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgges.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgges.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
   *                         SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
   *                         LWORK, RWORK, BWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBVSL, JOBVSR, SORT
   *       INTEGER            INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
   *       ..
   *       .. Array Arguments ..
   *       LOGICAL            BWORK( * )
   *       DOUBLE PRECISION   RWORK( * )
   *       COMPLEX*16         A( LDA, * ), ALPHA( * ), B( LDB, * ),
   *      $                   BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
   *      $                   WORK( * )
   *       ..
   *       .. Function Arguments ..
   *       LOGICAL            SELCTG
   *       EXTERNAL           SELCTG
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZGGES computes for a pair of N-by-N complex nonsymmetric matrices
   *> (A,B), the generalized eigenvalues, the generalized complex Schur
   *> form (S, T), and optionally left and/or right 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. The leading
   *> columns of VSL and VSR then form an unitary basis for the
   *> corresponding left and right eigenspaces (deflating subspaces).
   *>
   *> (If only the generalized eigenvalues are needed, use the driver
   *> ZGGEV instead, which is faster.)
   *>
   *> 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, and even for both being zero.
   *>
   *> A pair of matrices (S,T) is in generalized complex Schur form if S
   *> and T are upper triangular and, in addition, the diagonal elements
   *> of T are non-negative real numbers.
   *> \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 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.
   *>          An eigenvalue ALPHA(j)/BETA(j) is selected if
   *>          SELCTG(ALPHA(j),BETA(j)) is true.
   *>
   *>          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+2 (See INFO below).
   *> \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), j=1,...,N  and  BETA(j),
   *>          j=1,...,N  are the diagonals of the complex Schur form (A,B)
   *>          output by ZGGES. The  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] 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.  LWORK >= max(1,2*N).
   *>          For good performance, LWORK must generally be larger.
   *>
   *>          If LWORK = -1, then a workspace query is assumed; the routine
   *>          only calculates the optimal size of the WORK array, returns
   *>          this value as the first entry of the WORK array, and no error
   *>          message related to LWORK is issued by XERBLA.
   *> \endverbatim
   *>
   *> \param[out] RWORK
   *> \verbatim
   *>          RWORK is DOUBLE PRECISION array, dimension (8*N)
   *> \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 falied in ZTGSEN.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup complex16GEeigen
   *
   *  =====================================================================
       SUBROUTINE ZGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,        SUBROUTINE ZGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
      $                  SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK,       $                  SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
      $                  LWORK, RWORK, BWORK, INFO )       $                  LWORK, RWORK, BWORK, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2) --  *  -- LAPACK driver routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     November 2006  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOBVSL, JOBVSR, SORT        CHARACTER          JOBVSL, JOBVSR, SORT
Line 23 Line 291
       EXTERNAL           SELCTG        EXTERNAL           SELCTG
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZGGES computes for a pair of N-by-N complex nonsymmetric matrices  
 *  (A,B), the generalized eigenvalues, the generalized complex Schur  
 *  form (S, T), and optionally left and/or right 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. The leading  
 *  columns of VSL and VSR then form an unitary basis for the  
 *  corresponding left and right eigenspaces (deflating subspaces).  
 *  
 *  (If only the generalized eigenvalues are needed, use the driver  
 *  ZGGEV instead, which is faster.)  
 *  
 *  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, and even for both being zero.  
 *  
 *  A pair of matrices (S,T) is in generalized complex Schur form if S  
 *  and T are upper triangular and, in addition, the diagonal elements  
 *  of T are non-negative real numbers.  
 *  
 *  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.  
 *          An eigenvalue ALPHA(j)/BETA(j) is selected if  
 *          SELCTG(ALPHA(j),BETA(j)) is true.  
 *  
 *          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+2 (See INFO below).  
 *  
 *  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), j=1,...,N  and  BETA(j),  
 *          j=1,...,N  are the diagonals of the complex Schur form (A,B)  
 *          output by ZGGES. The  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.  
 *  
 *  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.  LWORK >= max(1,2*N).  
 *          For good performance, LWORK must generally be larger.  
 *  
 *          If LWORK = -1, then a workspace query is assumed; the routine  
 *          only calculates the optimal size of the WORK array, returns  
 *          this value as the first entry of the WORK array, and no error  
 *          message related to LWORK is issued by XERBLA.  
 *  
 *  RWORK   (workspace) DOUBLE PRECISION array, dimension (8*N)  
 *  
 *  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 falied in ZTGSEN.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.7  
changed lines
  Added in v.1.8

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