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version 1.1.1.1, 2010/01/26 15:22:46 version 1.17, 2023/08/07 08:38:48
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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 DGEGV + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegv.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgegv.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgegv.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
   *                         BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBVL, JOBVR
   *       INTEGER            INFO, LDA, LDB, LDVL, LDVR, LWORK, N
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
   *      $                   B( LDB, * ), BETA( * ), VL( LDVL, * ),
   *      $                   VR( LDVR, * ), WORK( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> This routine is deprecated and has been replaced by routine DGGEV.
   *>
   *> DGEGV computes the eigenvalues and, optionally, the left and/or right
   *> eigenvectors of a real matrix pair (A,B).
   *> Given two square matrices A and B,
   *> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
   *> eigenvalues lambda and corresponding (non-zero) eigenvectors x such
   *> that
   *>
   *>    A*x = lambda*B*x.
   *>
   *> An alternate form is to find the eigenvalues mu and corresponding
   *> eigenvectors y such that
   *>
   *>    mu*A*y = B*y.
   *>
   *> These two forms are equivalent with mu = 1/lambda and x = y if
   *> neither lambda nor mu is zero.  In order to deal with the case that
   *> lambda or mu is zero or small, two values alpha and beta are returned
   *> for each eigenvalue, such that lambda = alpha/beta and
   *> mu = beta/alpha.
   *>
   *> The vectors x and y in the above equations are right eigenvectors of
   *> the matrix pair (A,B).  Vectors u and v satisfying
   *>
   *>    u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B
   *>
   *> are left eigenvectors of (A,B).
   *>
   *> Note: this routine performs "full balancing" on A and B
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOBVL
   *> \verbatim
   *>          JOBVL is CHARACTER*1
   *>          = 'N':  do not compute the left generalized eigenvectors;
   *>          = 'V':  compute the left generalized eigenvectors (returned
   *>                  in VL).
   *> \endverbatim
   *>
   *> \param[in] JOBVR
   *> \verbatim
   *>          JOBVR is CHARACTER*1
   *>          = 'N':  do not compute the right generalized eigenvectors;
   *>          = 'V':  compute the right generalized eigenvectors (returned
   *>                  in VR).
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrices A, B, VL, and VR.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA, N)
   *>          On entry, the matrix A.
   *>          If JOBVL = 'V' or JOBVR = 'V', then on exit A
   *>          contains the real Schur form of A from the generalized Schur
   *>          factorization of the pair (A,B) after balancing.
   *>          If no eigenvectors were computed, then only the diagonal
   *>          blocks from the Schur form will be correct.  See DGGHRD and
   *>          DHGEQZ for details.
   *> \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.
   *>          If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
   *>          upper triangular matrix obtained from B in the generalized
   *>          Schur factorization of the pair (A,B) after balancing.
   *>          If no eigenvectors were computed, then only those elements of
   *>          B corresponding to the diagonal blocks from the Schur form of
   *>          A will be correct.  See DGGHRD and DHGEQZ for details.
   *> \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] VL
   *> \verbatim
   *>          VL is DOUBLE PRECISION array, dimension (LDVL,N)
   *>          If JOBVL = 'V', the left eigenvectors u(j) are stored
   *>          in the columns of VL, in the same order as their eigenvalues.
   *>          If the j-th eigenvalue is real, then u(j) = VL(:,j).
   *>          If the j-th and (j+1)-st eigenvalues form a complex conjugate
   *>          pair, then
   *>             u(j) = VL(:,j) + i*VL(:,j+1)
   *>          and
   *>            u(j+1) = VL(:,j) - i*VL(:,j+1).
   *>
   *>          Each eigenvector is scaled so that its largest component has
   *>          abs(real part) + abs(imag. part) = 1, except for eigenvectors
   *>          corresponding to an eigenvalue with alpha = beta = 0, which
   *>          are set to zero.
   *>          Not referenced if JOBVL = 'N'.
   *> \endverbatim
   *>
   *> \param[in] LDVL
   *> \verbatim
   *>          LDVL is INTEGER
   *>          The leading dimension of the matrix VL. LDVL >= 1, and
   *>          if JOBVL = 'V', LDVL >= N.
   *> \endverbatim
   *>
   *> \param[out] VR
   *> \verbatim
   *>          VR is DOUBLE PRECISION array, dimension (LDVR,N)
   *>          If JOBVR = 'V', the right eigenvectors x(j) are stored
   *>          in the columns of VR, in the same order as their eigenvalues.
   *>          If the j-th eigenvalue is real, then x(j) = VR(:,j).
   *>          If the j-th and (j+1)-st eigenvalues form a complex conjugate
   *>          pair, then
   *>            x(j) = VR(:,j) + i*VR(:,j+1)
   *>          and
   *>            x(j+1) = VR(:,j) - i*VR(:,j+1).
   *>
   *>          Each eigenvector is scaled so that its largest component has
   *>          abs(real part) + abs(imag. part) = 1, except for eigenvalues
   *>          corresponding to an eigenvalue with alpha = beta = 0, which
   *>          are set to zero.
   *>          Not referenced if JOBVR = 'N'.
   *> \endverbatim
   *>
   *> \param[in] LDVR
   *> \verbatim
   *>          LDVR is INTEGER
   *>          The leading dimension of the matrix VR. LDVR >= 1, and
   *>          if JOBVR = 'V', LDVR >= 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(1,8*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 + MAX( 6*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.  No eigenvectors have been
   *>                calculated, 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 DTGEVC
   *>                =N+8: error return from DGGBAK (computing VL)
   *>                =N+9: error return from DGGBAK (computing VR)
   *>                =N+10: error return from DLASCL (various calls)
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \ingroup doubleGEeigen
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  Balancing
   *>  ---------
   *>
   *>  This driver calls DGGBAL to both permute and scale rows and columns
   *>  of A and B.  The permutations PL and PR are chosen so that PL*A*PR
   *>  and PL*B*R will be upper triangular except for the diagonal blocks
   *>  A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
   *>  possible.  The diagonal scaling matrices DL and DR are chosen so
   *>  that the pair  DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
   *>  one (except for the elements that start out zero.)
   *>
   *>  After the eigenvalues and eigenvectors of the balanced matrices
   *>  have been computed, DGGBAK transforms the eigenvectors back to what
   *>  they would have been (in perfect arithmetic) if they had not been
   *>  balanced.
   *>
   *>  Contents of A and B on Exit
   *>  -------- -- - --- - -- ----
   *>
   *>  If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
   *>  both), then on exit the arrays A and B will contain the real Schur
   *>  form[*] of the "balanced" versions of A and B.  If no eigenvectors
   *>  are computed, then only the diagonal blocks will be correct.
   *>
   *>  [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",
   *>      by Golub & van Loan, pub. by Johns Hopkins U. Press.
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,        SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
      $                  BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )       $                  BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2) --  *  -- LAPACK driver routine --
 *  -- 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  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOBVL, JOBVR        CHARACTER          JOBVL, JOBVR
Line 16 Line 318
      $                   VR( LDVR, * ), WORK( * )       $                   VR( LDVR, * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  This routine is deprecated and has been replaced by routine DGGEV.  
 *  
 *  DGEGV computes the eigenvalues and, optionally, the left and/or right  
 *  eigenvectors of a real matrix pair (A,B).  
 *  Given two square matrices A and B,  
 *  the generalized nonsymmetric eigenvalue problem (GNEP) is to find the  
 *  eigenvalues lambda and corresponding (non-zero) eigenvectors x such  
 *  that  
 *  
 *     A*x = lambda*B*x.  
 *  
 *  An alternate form is to find the eigenvalues mu and corresponding  
 *  eigenvectors y such that  
 *  
 *     mu*A*y = B*y.  
 *  
 *  These two forms are equivalent with mu = 1/lambda and x = y if  
 *  neither lambda nor mu is zero.  In order to deal with the case that  
 *  lambda or mu is zero or small, two values alpha and beta are returned  
 *  for each eigenvalue, such that lambda = alpha/beta and  
 *  mu = beta/alpha.  
 *  
 *  The vectors x and y in the above equations are right eigenvectors of  
 *  the matrix pair (A,B).  Vectors u and v satisfying  
 *  
 *     u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B  
 *  
 *  are left eigenvectors of (A,B).  
 *  
 *  Note: this routine performs "full balancing" on A and B -- see  
 *  "Further Details", below.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  JOBVL   (input) CHARACTER*1  
 *          = 'N':  do not compute the left generalized eigenvectors;  
 *          = 'V':  compute the left generalized eigenvectors (returned  
 *                  in VL).  
 *  
 *  JOBVR   (input) CHARACTER*1  
 *          = 'N':  do not compute the right generalized eigenvectors;  
 *          = 'V':  compute the right generalized eigenvectors (returned  
 *                  in VR).  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrices A, B, VL, and VR.  N >= 0.  
 *  
 *  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)  
 *          On entry, the matrix A.  
 *          If JOBVL = 'V' or JOBVR = 'V', then on exit A  
 *          contains the real Schur form of A from the generalized Schur  
 *          factorization of the pair (A,B) after balancing.  
 *          If no eigenvectors were computed, then only the diagonal  
 *          blocks from the Schur form will be correct.  See DGGHRD and  
 *          DHGEQZ for details.  
 *  
 *  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.  
 *          If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the  
 *          upper triangular matrix obtained from B in the generalized  
 *          Schur factorization of the pair (A,B) after balancing.  
 *          If no eigenvectors were computed, then only those elements of  
 *          B corresponding to the diagonal blocks from the Schur form of  
 *          A will be correct.  See DGGHRD and DHGEQZ for details.  
 *  
 *  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.  
 *  
 *  VL      (output) DOUBLE PRECISION array, dimension (LDVL,N)  
 *          If JOBVL = 'V', the left eigenvectors u(j) are stored  
 *          in the columns of VL, in the same order as their eigenvalues.  
 *          If the j-th eigenvalue is real, then u(j) = VL(:,j).  
 *          If the j-th and (j+1)-st eigenvalues form a complex conjugate  
 *          pair, then  
 *             u(j) = VL(:,j) + i*VL(:,j+1)  
 *          and  
 *            u(j+1) = VL(:,j) - i*VL(:,j+1).  
 *  
 *          Each eigenvector is scaled so that its largest component has  
 *          abs(real part) + abs(imag. part) = 1, except for eigenvectors  
 *          corresponding to an eigenvalue with alpha = beta = 0, which  
 *          are set to zero.  
 *          Not referenced if JOBVL = 'N'.  
 *  
 *  LDVL    (input) INTEGER  
 *          The leading dimension of the matrix VL. LDVL >= 1, and  
 *          if JOBVL = 'V', LDVL >= N.  
 *  
 *  VR      (output) DOUBLE PRECISION array, dimension (LDVR,N)  
 *          If JOBVR = 'V', the right eigenvectors x(j) are stored  
 *          in the columns of VR, in the same order as their eigenvalues.  
 *          If the j-th eigenvalue is real, then x(j) = VR(:,j).  
 *          If the j-th and (j+1)-st eigenvalues form a complex conjugate  
 *          pair, then  
 *            x(j) = VR(:,j) + i*VR(:,j+1)  
 *          and  
 *            x(j+1) = VR(:,j) - i*VR(:,j+1).  
 *  
 *          Each eigenvector is scaled so that its largest component has  
 *          abs(real part) + abs(imag. part) = 1, except for eigenvalues  
 *          corresponding to an eigenvalue with alpha = beta = 0, which  
 *          are set to zero.  
 *          Not referenced if JOBVR = 'N'.  
 *  
 *  LDVR    (input) INTEGER  
 *          The leading dimension of the matrix VR. LDVR >= 1, and  
 *          if JOBVR = 'V', LDVR >= 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(1,8*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 + MAX( 6*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.  No eigenvectors have been  
 *                calculated, 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 DTGEVC  
 *                =N+8: error return from DGGBAK (computing VL)  
 *                =N+9: error return from DGGBAK (computing VR)  
 *                =N+10: error return from DLASCL (various calls)  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  Balancing  
 *  ---------  
 *  
 *  This driver calls DGGBAL to both permute and scale rows and columns  
 *  of A and B.  The permutations PL and PR are chosen so that PL*A*PR  
 *  and PL*B*R will be upper triangular except for the diagonal blocks  
 *  A(i:j,i:j) and B(i:j,i:j), with i and j as close together as  
 *  possible.  The diagonal scaling matrices DL and DR are chosen so  
 *  that the pair  DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to  
 *  one (except for the elements that start out zero.)  
 *  
 *  After the eigenvalues and eigenvectors of the balanced matrices  
 *  have been computed, DGGBAK transforms the eigenvectors back to what  
 *  they would have been (in perfect arithmetic) if they had not been  
 *  balanced.  
 *  
 *  Contents of A and B on Exit  
 *  -------- -- - --- - -- ----  
 *  
 *  If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or  
 *  both), then on exit the arrays A and B will contain the real Schur  
 *  form[*] of the "balanced" versions of A and B.  If no eigenvectors  
 *  are computed, then only the diagonal blocks will be correct.  
 *  
 *  [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",  
 *      by Golub & van Loan, pub. by Johns Hopkins U. Press.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.1.1.1  
changed lines
  Added in v.1.17

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