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   *> \brief <b> DGGEVX 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 DGGEVX + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggevx.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggevx.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggevx.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
   *                          ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
   *                          IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
   *                          RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       CHARACTER          BALANC, JOBVL, JOBVR, SENSE
   *       INTEGER            IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
   *       DOUBLE PRECISION   ABNRM, BBNRM
   *       ..
   *       .. Array Arguments ..
   *       LOGICAL            BWORK( * )
   *       INTEGER            IWORK( * )
   *       DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
   *      $                   B( LDB, * ), BETA( * ), LSCALE( * ),
   *      $                   RCONDE( * ), RCONDV( * ), RSCALE( * ),
   *      $                   VL( LDVL, * ), VR( LDVR, * ), WORK( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
   *> the generalized eigenvalues, and optionally, the left and/or right
   *> generalized eigenvectors.
   *>
   *> Optionally also, it computes a balancing transformation to improve
   *> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
   *> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
   *> the eigenvalues (RCONDE), and reciprocal condition numbers for the
   *> right eigenvectors (RCONDV).
   *>
   *> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
   *> lambda or a ratio alpha/beta = lambda, such that A - lambda*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.
   *>
   *> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
   *> of (A,B) satisfies
   *>
   *>                  A * v(j) = lambda(j) * B * v(j) .
   *>
   *> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
   *> of (A,B) satisfies
   *>
   *>                  u(j)**H * A  = lambda(j) * u(j)**H * B.
   *>
   *> where u(j)**H is the conjugate-transpose of u(j).
   *>
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] BALANC
   *> \verbatim
   *>          BALANC is CHARACTER*1
   *>          Specifies the balance option to be performed.
   *>          = 'N':  do not diagonally scale or permute;
   *>          = 'P':  permute only;
   *>          = 'S':  scale only;
   *>          = 'B':  both permute and scale.
   *>          Computed reciprocal condition numbers will be for the
   *>          matrices after permuting and/or balancing. Permuting does
   *>          not change condition numbers (in exact arithmetic), but
   *>          balancing does.
   *> \endverbatim
   *>
   *> \param[in] JOBVL
   *> \verbatim
   *>          JOBVL is CHARACTER*1
   *>          = 'N':  do not compute the left generalized eigenvectors;
   *>          = 'V':  compute the left generalized eigenvectors.
   *> \endverbatim
   *>
   *> \param[in] JOBVR
   *> \verbatim
   *>          JOBVR is CHARACTER*1
   *>          = 'N':  do not compute the right generalized eigenvectors;
   *>          = 'V':  compute the right generalized eigenvectors.
   *> \endverbatim
   *>
   *> \param[in] SENSE
   *> \verbatim
   *>          SENSE is CHARACTER*1
   *>          Determines which reciprocal condition numbers are computed.
   *>          = 'N': none are computed;
   *>          = 'E': computed for eigenvalues only;
   *>          = 'V': computed for eigenvectors only;
   *>          = 'B': computed for eigenvalues and eigenvectors.
   *> \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 in the pair (A,B).
   *>          On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
   *>          or both, then A contains the first part of the real Schur
   *>          form of the "balanced" versions of the input A and B.
   *> \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 in the pair (A,B).
   *>          On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
   *>          or both, then B contains the second part of the real Schur
   *>          form of the "balanced" versions of the input A and B.
   *> \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)
   *> \endverbatim
   *>
   *> \param[out] ALPHAI
   *> \verbatim
   *>          ALPHAI is DOUBLE PRECISION array, dimension (N)
   *> \endverbatim
   *>
   *> \param[out] BETA
   *> \verbatim
   *>          BETA is DOUBLE PRECISION array, dimension (N)
   *>          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
   *>          be the generalized eigenvalues.  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) negative.
   *>
   *>          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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] VL
   *> \verbatim
   *>          VL is DOUBLE PRECISION array, dimension (LDVL,N)
   *>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
   *>          after another in the columns of VL, in the same order as
   *>          their eigenvalues. If the j-th eigenvalue is real, then
   *>          u(j) = VL(:,j), the j-th column of VL. If the j-th and
   *>          (j+1)-th 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 will be scaled so the largest component have
   *>          abs(real part) + abs(imag. part) = 1.
   *>          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 v(j) are stored one
   *>          after another in the columns of VR, in the same order as
   *>          their eigenvalues. If the j-th eigenvalue is real, then
   *>          v(j) = VR(:,j), the j-th column of VR. If the j-th and
   *>          (j+1)-th eigenvalues form a complex conjugate pair, then
   *>          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
   *>          Each eigenvector will be scaled so the largest component have
   *>          abs(real part) + abs(imag. part) = 1.
   *>          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] ILO
   *> \verbatim
   *>          ILO is INTEGER
   *> \endverbatim
   *>
   *> \param[out] IHI
   *> \verbatim
   *>          IHI is INTEGER
   *>          ILO and IHI are integer values such that on exit
   *>          A(i,j) = 0 and B(i,j) = 0 if i > j and
   *>          j = 1,...,ILO-1 or i = IHI+1,...,N.
   *>          If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
   *> \endverbatim
   *>
   *> \param[out] LSCALE
   *> \verbatim
   *>          LSCALE is DOUBLE PRECISION array, dimension (N)
   *>          Details of the permutations and scaling factors applied
   *>          to the left side of A and B.  If PL(j) is the index of the
   *>          row interchanged with row j, and DL(j) is the scaling
   *>          factor applied to row j, then
   *>            LSCALE(j) = PL(j)  for j = 1,...,ILO-1
   *>                      = DL(j)  for j = ILO,...,IHI
   *>                      = PL(j)  for j = IHI+1,...,N.
   *>          The order in which the interchanges are made is N to IHI+1,
   *>          then 1 to ILO-1.
   *> \endverbatim
   *>
   *> \param[out] RSCALE
   *> \verbatim
   *>          RSCALE is DOUBLE PRECISION array, dimension (N)
   *>          Details of the permutations and scaling factors applied
   *>          to the right side of A and B.  If PR(j) is the index of the
   *>          column interchanged with column j, and DR(j) is the scaling
   *>          factor applied to column j, then
   *>            RSCALE(j) = PR(j)  for j = 1,...,ILO-1
   *>                      = DR(j)  for j = ILO,...,IHI
   *>                      = PR(j)  for j = IHI+1,...,N
   *>          The order in which the interchanges are made is N to IHI+1,
   *>          then 1 to ILO-1.
   *> \endverbatim
   *>
   *> \param[out] ABNRM
   *> \verbatim
   *>          ABNRM is DOUBLE PRECISION
   *>          The one-norm of the balanced matrix A.
   *> \endverbatim
   *>
   *> \param[out] BBNRM
   *> \verbatim
   *>          BBNRM is DOUBLE PRECISION
   *>          The one-norm of the balanced matrix B.
   *> \endverbatim
   *>
   *> \param[out] RCONDE
   *> \verbatim
   *>          RCONDE is DOUBLE PRECISION array, dimension (N)
   *>          If SENSE = 'E' or 'B', the reciprocal condition numbers of
   *>          the eigenvalues, stored in consecutive elements of the array.
   *>          For a complex conjugate pair of eigenvalues two consecutive
   *>          elements of RCONDE are set to the same value. Thus RCONDE(j),
   *>          RCONDV(j), and the j-th columns of VL and VR all correspond
   *>          to the j-th eigenpair.
   *>          If SENSE = 'N or 'V', RCONDE is not referenced.
   *> \endverbatim
   *>
   *> \param[out] RCONDV
   *> \verbatim
   *>          RCONDV is DOUBLE PRECISION array, dimension (N)
   *>          If SENSE = 'V' or 'B', the estimated reciprocal condition
   *>          numbers of the eigenvectors, stored in consecutive elements
   *>          of the array. For a complex eigenvector two consecutive
   *>          elements of RCONDV are set to the same value. If the
   *>          eigenvalues cannot be reordered to compute RCONDV(j),
   *>          RCONDV(j) is set to 0; this can only occur when the true
   *>          value would be very small anyway.
   *>          If SENSE = 'N' or 'E', RCONDV is not referenced.
   *> \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(1,2*N).
   *>          If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
   *>          LWORK >= max(1,6*N).
   *>          If SENSE = 'E' or 'B', LWORK >= max(1,10*N).
   *>          If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
   *>
   *>          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] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (N+6)
   *>          If SENSE = 'E', IWORK is not referenced.
   *> \endverbatim
   *>
   *> \param[out] BWORK
   *> \verbatim
   *>          BWORK is LOGICAL array, dimension (N)
   *>          If SENSE = 'N', BWORK is not referenced.
   *> \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:  =N+1: other than QZ iteration failed in DHGEQZ.
   *>                =N+2: error return from DTGEVC.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \date April 2012
   *
   *> \ingroup doubleGEeigen
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  Balancing a matrix pair (A,B) includes, first, permuting rows and
   *>  columns to isolate eigenvalues, second, applying diagonal similarity
   *>  transformation to the rows and columns to make the rows and columns
   *>  as close in norm as possible. The computed reciprocal condition
   *>  numbers correspond to the balanced matrix. Permuting rows and columns
   *>  will not change the condition numbers (in exact arithmetic) but
   *>  diagonal scaling will.  For further explanation of balancing, see
   *>  section 4.11.1.2 of LAPACK Users' Guide.
   *>
   *>  An approximate error bound on the chordal distance between the i-th
   *>  computed generalized eigenvalue w and the corresponding exact
   *>  eigenvalue lambda is
   *>
   *>       chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
   *>
   *>  An approximate error bound for the angle between the i-th computed
   *>  eigenvector VL(i) or VR(i) is given by
   *>
   *>       EPS * norm(ABNRM, BBNRM) / DIF(i).
   *>
   *>  For further explanation of the reciprocal condition numbers RCONDE
   *>  and RCONDV, see section 4.11 of LAPACK User's Guide.
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,        SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
      $                   ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,       $                   ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
      $                   IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,       $                   IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
      $                   RCONDV, WORK, LWORK, IWORK, BWORK, INFO )       $                   RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2) --  *  -- LAPACK driver routine (version 3.7.0) --
 *  -- 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  *     April 2012
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          BALANC, JOBVL, JOBVR, SENSE        CHARACTER          BALANC, JOBVL, JOBVR, SENSE
Line 22 Line 410
      $                   VL( LDVL, * ), VR( LDVR, * ), WORK( * )       $                   VL( LDVL, * ), VR( LDVR, * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)  
 *  the generalized eigenvalues, and optionally, the left and/or right  
 *  generalized eigenvectors.  
 *  
 *  Optionally also, it computes a balancing transformation to improve  
 *  the conditioning of the eigenvalues and eigenvectors (ILO, IHI,  
 *  LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for  
 *  the eigenvalues (RCONDE), and reciprocal condition numbers for the  
 *  right eigenvectors (RCONDV).  
 *  
 *  A generalized eigenvalue for a pair of matrices (A,B) is a scalar  
 *  lambda or a ratio alpha/beta = lambda, such that A - lambda*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.  
 *  
 *  The right eigenvector v(j) corresponding to the eigenvalue lambda(j)  
 *  of (A,B) satisfies  
 *  
 *                   A * v(j) = lambda(j) * B * v(j) .  
 *  
 *  The left eigenvector u(j) corresponding to the eigenvalue lambda(j)  
 *  of (A,B) satisfies  
 *  
 *                   u(j)**H * A  = lambda(j) * u(j)**H * B.  
 *  
 *  where u(j)**H is the conjugate-transpose of u(j).  
 *  
 *  
 *  Arguments  
 *  =========  
 *  
 *  BALANC  (input) CHARACTER*1  
 *          Specifies the balance option to be performed.  
 *          = 'N':  do not diagonally scale or permute;  
 *          = 'P':  permute only;  
 *          = 'S':  scale only;  
 *          = 'B':  both permute and scale.  
 *          Computed reciprocal condition numbers will be for the  
 *          matrices after permuting and/or balancing. Permuting does  
 *          not change condition numbers (in exact arithmetic), but  
 *          balancing does.  
 *  
 *  JOBVL   (input) CHARACTER*1  
 *          = 'N':  do not compute the left generalized eigenvectors;  
 *          = 'V':  compute the left generalized eigenvectors.  
 *  
 *  JOBVR   (input) CHARACTER*1  
 *          = 'N':  do not compute the right generalized eigenvectors;  
 *          = 'V':  compute the right generalized eigenvectors.  
 *  
 *  SENSE   (input) CHARACTER*1  
 *          Determines which reciprocal condition numbers are computed.  
 *          = 'N': none are computed;  
 *          = 'E': computed for eigenvalues only;  
 *          = 'V': computed for eigenvectors only;  
 *          = 'B': computed for eigenvalues and eigenvectors.  
 *  
 *  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 in the pair (A,B).  
 *          On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'  
 *          or both, then A contains the first part of the real Schur  
 *          form of the "balanced" versions of the input A and B.  
 *  
 *  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 in the pair (A,B).  
 *          On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'  
 *          or both, then B contains the second part of the real Schur  
 *          form of the "balanced" versions of the input A and B.  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of B.  LDB >= max(1,N).  
 *  
 *  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)  
 *  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)  
 *  BETA    (output) DOUBLE PRECISION array, dimension (N)  
 *          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will  
 *          be the generalized eigenvalues.  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) negative.  
 *  
 *          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI will be always less  
 *          than and usually comparable with norm(A) in magnitude, and  
 *          BETA always less than and usually comparable with norm(B).  
 *  
 *  VL      (output) DOUBLE PRECISION array, dimension (LDVL,N)  
 *          If JOBVL = 'V', the left eigenvectors u(j) are stored one  
 *          after another in the columns of VL, in the same order as  
 *          their eigenvalues. If the j-th eigenvalue is real, then  
 *          u(j) = VL(:,j), the j-th column of VL. If the j-th and  
 *          (j+1)-th 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 will be scaled so the largest component have  
 *          abs(real part) + abs(imag. part) = 1.  
 *          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 v(j) are stored one  
 *          after another in the columns of VR, in the same order as  
 *          their eigenvalues. If the j-th eigenvalue is real, then  
 *          v(j) = VR(:,j), the j-th column of VR. If the j-th and  
 *          (j+1)-th eigenvalues form a complex conjugate pair, then  
 *          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).  
 *          Each eigenvector will be scaled so the largest component have  
 *          abs(real part) + abs(imag. part) = 1.  
 *          Not referenced if JOBVR = 'N'.  
 *  
 *  LDVR    (input) INTEGER  
 *          The leading dimension of the matrix VR. LDVR >= 1, and  
 *          if JOBVR = 'V', LDVR >= N.  
 *  
 *  ILO     (output) INTEGER  
 *  IHI     (output) INTEGER  
 *          ILO and IHI are integer values such that on exit  
 *          A(i,j) = 0 and B(i,j) = 0 if i > j and  
 *          j = 1,...,ILO-1 or i = IHI+1,...,N.  
 *          If BALANC = 'N' or 'S', ILO = 1 and IHI = N.  
 *  
 *  LSCALE  (output) DOUBLE PRECISION array, dimension (N)  
 *          Details of the permutations and scaling factors applied  
 *          to the left side of A and B.  If PL(j) is the index of the  
 *          row interchanged with row j, and DL(j) is the scaling  
 *          factor applied to row j, then  
 *            LSCALE(j) = PL(j)  for j = 1,...,ILO-1  
 *                      = DL(j)  for j = ILO,...,IHI  
 *                      = PL(j)  for j = IHI+1,...,N.  
 *          The order in which the interchanges are made is N to IHI+1,  
 *          then 1 to ILO-1.  
 *  
 *  RSCALE  (output) DOUBLE PRECISION array, dimension (N)  
 *          Details of the permutations and scaling factors applied  
 *          to the right side of A and B.  If PR(j) is the index of the  
 *          column interchanged with column j, and DR(j) is the scaling  
 *          factor applied to column j, then  
 *            RSCALE(j) = PR(j)  for j = 1,...,ILO-1  
 *                      = DR(j)  for j = ILO,...,IHI  
 *                      = PR(j)  for j = IHI+1,...,N  
 *          The order in which the interchanges are made is N to IHI+1,  
 *          then 1 to ILO-1.  
 *  
 *  ABNRM   (output) DOUBLE PRECISION  
 *          The one-norm of the balanced matrix A.  
 *  
 *  BBNRM   (output) DOUBLE PRECISION  
 *          The one-norm of the balanced matrix B.  
 *  
 *  RCONDE  (output) DOUBLE PRECISION array, dimension (N)  
 *          If SENSE = 'E' or 'B', the reciprocal condition numbers of  
 *          the eigenvalues, stored in consecutive elements of the array.  
 *          For a complex conjugate pair of eigenvalues two consecutive  
 *          elements of RCONDE are set to the same value. Thus RCONDE(j),  
 *          RCONDV(j), and the j-th columns of VL and VR all correspond  
 *          to the j-th eigenpair.  
 *          If SENSE = 'N or 'V', RCONDE is not referenced.  
 *  
 *  RCONDV  (output) DOUBLE PRECISION array, dimension (N)  
 *          If SENSE = 'V' or 'B', the estimated reciprocal condition  
 *          numbers of the eigenvectors, stored in consecutive elements  
 *          of the array. For a complex eigenvector two consecutive  
 *          elements of RCONDV are set to the same value. If the  
 *          eigenvalues cannot be reordered to compute RCONDV(j),  
 *          RCONDV(j) is set to 0; this can only occur when the true  
 *          value would be very small anyway.  
 *          If SENSE = 'N' or 'E', RCONDV is not referenced.  
 *  
 *  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(1,2*N).  
 *          If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',  
 *          LWORK >= max(1,6*N).  
 *          If SENSE = 'E' or 'B', LWORK >= max(1,10*N).  
 *          If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.  
 *  
 *          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.  
 *  
 *  IWORK   (workspace) INTEGER array, dimension (N+6)  
 *          If SENSE = 'E', IWORK is not referenced.  
 *  
 *  BWORK   (workspace) LOGICAL array, dimension (N)  
 *          If SENSE = 'N', BWORK is not referenced.  
 *  
 *  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:  =N+1: other than QZ iteration failed in DHGEQZ.  
 *                =N+2: error return from DTGEVC.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  Balancing a matrix pair (A,B) includes, first, permuting rows and  
 *  columns to isolate eigenvalues, second, applying diagonal similarity  
 *  transformation to the rows and columns to make the rows and columns  
 *  as close in norm as possible. The computed reciprocal condition  
 *  numbers correspond to the balanced matrix. Permuting rows and columns  
 *  will not change the condition numbers (in exact arithmetic) but  
 *  diagonal scaling will.  For further explanation of balancing, see  
 *  section 4.11.1.2 of LAPACK Users' Guide.  
 *  
 *  An approximate error bound on the chordal distance between the i-th  
 *  computed generalized eigenvalue w and the corresponding exact  
 *  eigenvalue lambda is  
 *  
 *       chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)  
 *  
 *  An approximate error bound for the angle between the i-th computed  
 *  eigenvector VL(i) or VR(i) is given by  
 *  
 *       EPS * norm(ABNRM, BBNRM) / DIF(i).  
 *  
 *  For further explanation of the reciprocal condition numbers RCONDE  
 *  and RCONDV, see section 4.11 of LAPACK User's Guide.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Line 283 Line 432
 *     .. External Subroutines ..  *     .. External Subroutines ..
       EXTERNAL           DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD,        EXTERNAL           DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD,
      $                   DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGEVC,       $                   DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGEVC,
      $                   DTGSNA, XERBLA        $                   DTGSNA, XERBLA
 *     ..  *     ..
 *     .. External Functions ..  *     .. External Functions ..
       LOGICAL            LSAME        LOGICAL            LSAME
Line 700 Line 849
 *  *
 *     Undo scaling if necessary  *     Undo scaling if necessary
 *  *
     130 CONTINUE
   *
       IF( ILASCL ) THEN        IF( ILASCL ) THEN
          CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )           CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
          CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )           CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
Line 709 Line 860
          CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )           CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
       END IF        END IF
 *  *
   130 CONTINUE  
       WORK( 1 ) = MAXWRK        WORK( 1 ) = MAXWRK
 *  
       RETURN        RETURN
 *  *
 *     End of DGGEVX  *     End of DGGEVX
Removed from v.1.6  
changed lines
  Added in v.1.16

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