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version 1.7, 2010/12/21 13:53:25 version 1.8, 2011/11/21 20:42:50
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   *> \brief <b> DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DGEGS + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegs.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgegs.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgegs.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
   *                         ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
   *                         LWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBVSL, JOBVSR
   *       INTEGER            INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
   *      $                   B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
   *      $                   VSR( LDVSR, * ), WORK( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> This routine is deprecated and has been replaced by routine DGGES.
   *>
   *> DGEGS computes the eigenvalues, real Schur form, and, optionally,
   *> left and or/right Schur vectors of a real matrix pair (A,B).
   *> Given two square matrices A and B, the generalized real Schur
   *> factorization has the form
   *>
   *>   A = Q*S*Z**T,  B = Q*T*Z**T
   *>
   *> where Q and Z are orthogonal matrices, T is upper triangular, and S
   *> is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
   *> blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
   *> of eigenvalues of (A,B).  The columns of Q are the left Schur vectors
   *> and the columns of Z are the right Schur vectors.
   *>
   *> If only the eigenvalues of (A,B) are needed, the driver routine
   *> DGEGV should be used instead.  See DGEGV for a description of the
   *> eigenvalues of the generalized nonsymmetric eigenvalue problem
   *> (GNEP).
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOBVSL
   *> \verbatim
   *>          JOBVSL is CHARACTER*1
   *>          = 'N':  do not compute the left Schur vectors;
   *>          = 'V':  compute the left Schur vectors (returned in VSL).
   *> \endverbatim
   *>
   *> \param[in] JOBVSR
   *> \verbatim
   *>          JOBVSR is CHARACTER*1
   *>          = 'N':  do not compute the right Schur vectors;
   *>          = 'V':  compute the right Schur vectors (returned in VSR).
   *> \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 DOUBLE PRECISION array, dimension (LDA, N)
   *>          On entry, the matrix A.
   *>          On exit, the upper quasi-triangular matrix S from the
   *>          generalized real Schur factorization.
   *> \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 DOUBLE PRECISION array, dimension (LDB, N)
   *>          On entry, the matrix B.
   *>          On exit, the upper triangular matrix T from the generalized
   *>          real Schur factorization.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of B.  LDB >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] ALPHAR
   *> \verbatim
   *>          ALPHAR is DOUBLE PRECISION array, dimension (N)
   *>          The real parts of each scalar alpha defining an eigenvalue
   *>          of GNEP.
   *> \endverbatim
   *>
   *> \param[out] ALPHAI
   *> \verbatim
   *>          ALPHAI is DOUBLE PRECISION array, dimension (N)
   *>          The imaginary parts of each scalar alpha defining an
   *>          eigenvalue of GNEP.  If ALPHAI(j) is zero, then the j-th
   *>          eigenvalue is real; if positive, then the j-th and (j+1)-st
   *>          eigenvalues are a complex conjugate pair, with
   *>          ALPHAI(j+1) = -ALPHAI(j).
   *> \endverbatim
   *>
   *> \param[out] BETA
   *> \verbatim
   *>          BETA is DOUBLE PRECISION array, dimension (N)
   *>          The scalars beta that define the eigenvalues of GNEP.
   *>          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
   *>          beta = BETA(j) represent the j-th eigenvalue of the matrix
   *>          pair (A,B), in one of the forms lambda = alpha/beta or
   *>          mu = beta/alpha.  Since either lambda or mu may overflow,
   *>          they should not, in general, be computed.
   *> \endverbatim
   *>
   *> \param[out] VSL
   *> \verbatim
   *>          VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
   *>          If JOBVSL = 'V', the matrix of left Schur vectors Q.
   *>          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 DOUBLE PRECISION array, dimension (LDVSR,N)
   *>          If JOBVSR = 'V', the matrix of right Schur vectors Z.
   *>          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 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(1,4*N).
   *>          For good performance, LWORK must generally be larger.
   *>          To compute the optimal value of LWORK, call ILAENV to get
   *>          blocksizes (for DGEQRF, DORMQR, and DORGQR.)  Then compute:
   *>          NB  -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR
   *>          The optimal LWORK is  2*N + N*(NB+1).
   *>
   *>          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] 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 ALPHAR(j), ALPHAI(j), and BETA(j) should
   *>                be correct for j=INFO+1,...,N.
   *>          > N:  errors that usually indicate LAPACK problems:
   *>                =N+1: error return from DGGBAL
   *>                =N+2: error return from DGEQRF
   *>                =N+3: error return from DORMQR
   *>                =N+4: error return from DORGQR
   *>                =N+5: error return from DGGHRD
   *>                =N+6: error return from DHGEQZ (other than failed
   *>                                                iteration)
   *>                =N+7: error return from DGGBAK (computing VSL)
   *>                =N+8: error return from DGGBAK (computing VSR)
   *>                =N+9: error return from DLASCL (various places)
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleGEeigen
   *
   *  =====================================================================
       SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,        SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
      $                  ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,       $                  ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
      $                  LWORK, INFO )       $                  LWORK, 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        CHARACTER          JOBVSL, JOBVSR
Line 17 Line 242
      $                   VSR( LDVSR, * ), WORK( * )       $                   VSR( LDVSR, * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  This routine is deprecated and has been replaced by routine DGGES.  
 *  
 *  DGEGS computes the eigenvalues, real Schur form, and, optionally,  
 *  left and or/right Schur vectors of a real matrix pair (A,B).  
 *  Given two square matrices A and B, the generalized real Schur  
 *  factorization has the form  
 *  
 *    A = Q*S*Z**T,  B = Q*T*Z**T  
 *  
 *  where Q and Z are orthogonal matrices, T is upper triangular, and S  
 *  is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal  
 *  blocks, the 2-by-2 blocks corresponding to complex conjugate pairs  
 *  of eigenvalues of (A,B).  The columns of Q are the left Schur vectors  
 *  and the columns of Z are the right Schur vectors.  
 *  
 *  If only the eigenvalues of (A,B) are needed, the driver routine  
 *  DGEGV should be used instead.  See DGEGV for a description of the  
 *  eigenvalues of the generalized nonsymmetric eigenvalue problem  
 *  (GNEP).  
 *  
 *  Arguments  
 *  =========  
 *  
 *  JOBVSL  (input) CHARACTER*1  
 *          = 'N':  do not compute the left Schur vectors;  
 *          = 'V':  compute the left Schur vectors (returned in VSL).  
 *  
 *  JOBVSR  (input) CHARACTER*1  
 *          = 'N':  do not compute the right Schur vectors;  
 *          = 'V':  compute the right Schur vectors (returned in VSR).  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrices A, B, VSL, and VSR.  N >= 0.  
 *  
 *  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)  
 *          On entry, the matrix A.  
 *          On exit, the upper quasi-triangular matrix S from the  
 *          generalized real Schur factorization.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of A.  LDA >= max(1,N).  
 *  
 *  B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)  
 *          On entry, the matrix B.  
 *          On exit, the upper triangular matrix T from the generalized  
 *          real Schur factorization.  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of B.  LDB >= max(1,N).  
 *  
 *  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)  
 *          The real parts of each scalar alpha defining an eigenvalue  
 *          of GNEP.  
 *  
 *  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)  
 *          The imaginary parts of each scalar alpha defining an  
 *          eigenvalue of GNEP.  If ALPHAI(j) is zero, then the j-th  
 *          eigenvalue is real; if positive, then the j-th and (j+1)-st  
 *          eigenvalues are a complex conjugate pair, with  
 *          ALPHAI(j+1) = -ALPHAI(j).  
 *  
 *  BETA    (output) DOUBLE PRECISION array, dimension (N)  
 *          The scalars beta that define the eigenvalues of GNEP.  
 *          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and  
 *          beta = BETA(j) represent the j-th eigenvalue of the matrix  
 *          pair (A,B), in one of the forms lambda = alpha/beta or  
 *          mu = beta/alpha.  Since either lambda or mu may overflow,  
 *          they should not, in general, be computed.  
 *  
 *  VSL     (output) DOUBLE PRECISION array, dimension (LDVSL,N)  
 *          If JOBVSL = 'V', the matrix of left Schur vectors Q.  
 *          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) DOUBLE PRECISION array, dimension (LDVSR,N)  
 *          If JOBVSR = 'V', the matrix of right Schur vectors Z.  
 *          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) 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(1,4*N).  
 *          For good performance, LWORK must generally be larger.  
 *          To compute the optimal value of LWORK, call ILAENV to get  
 *          blocksizes (for DGEQRF, DORMQR, and DORGQR.)  Then compute:  
 *          NB  -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR  
 *          The optimal LWORK is  2*N + N*(NB+1).  
 *  
 *          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.  
 *  
 *  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 ALPHAR(j), ALPHAI(j), and BETA(j) should  
 *                be correct for j=INFO+1,...,N.  
 *          > N:  errors that usually indicate LAPACK problems:  
 *                =N+1: error return from DGGBAL  
 *                =N+2: error return from DGEQRF  
 *                =N+3: error return from DORMQR  
 *                =N+4: error return from DORGQR  
 *                =N+5: error return from DGGHRD  
 *                =N+6: error return from DHGEQZ (other than failed  
 *                                                iteration)  
 *                =N+7: error return from DGGBAK (computing VSL)  
 *                =N+8: error return from DGGBAK (computing VSR)  
 *                =N+9: error return from DLASCL (various places)  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.7  
changed lines
  Added in v.1.8

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