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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 DGEEVX + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeevx.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeevx.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeevx.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
   *                          VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
   *                          RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          BALANC, JOBVL, JOBVR, SENSE
   *       INTEGER            IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
   *       DOUBLE PRECISION   ABNRM
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IWORK( * )
   *       DOUBLE PRECISION   A( LDA, * ), RCONDE( * ), RCONDV( * ),
   *      $                   SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
   *      $                   WI( * ), WORK( * ), WR( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DGEEVX computes for an N-by-N real nonsymmetric matrix A, the
   *> eigenvalues and, optionally, the left and/or right eigenvectors.
   *>
   *> Optionally also, it computes a balancing transformation to improve
   *> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
   *> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
   *> (RCONDE), and reciprocal condition numbers for the right
   *> eigenvectors (RCONDV).
   *>
   *> The right eigenvector v(j) of A satisfies
   *>                  A * v(j) = lambda(j) * v(j)
   *> where lambda(j) is its eigenvalue.
   *> The left eigenvector u(j) of A satisfies
   *>               u(j)**T * A = lambda(j) * u(j)**T
   *> where u(j)**T denotes the transpose of u(j).
   *>
   *> The computed eigenvectors are normalized to have Euclidean norm
   *> equal to 1 and largest component real.
   *>
   *> Balancing a matrix means permuting the rows and columns to make it
   *> more nearly upper triangular, and applying a diagonal similarity
   *> transformation D * A * D**(-1), where D is a diagonal matrix, to
   *> make its rows and columns closer in norm and the condition numbers
   *> of its eigenvalues and eigenvectors smaller.  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 4.10.2 of the LAPACK
   *> Users' Guide.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] BALANC
   *> \verbatim
   *>          BALANC is CHARACTER*1
   *>          Indicates how the input matrix should be diagonally scaled
   *>          and/or permuted to improve the conditioning of its
   *>          eigenvalues.
   *>          = 'N': Do not diagonally scale or permute;
   *>          = 'P': Perform permutations to make the matrix more nearly
   *>                 upper triangular. Do not diagonally scale;
   *>          = 'S': Diagonally scale the matrix, i.e. replace A by
   *>                 D*A*D**(-1), where D is a diagonal matrix chosen
   *>                 to make the rows and columns of A more equal in
   *>                 norm. Do not permute;
   *>          = 'B': Both diagonally scale and permute A.
   *>
   *>          Computed reciprocal condition numbers will be for the matrix
   *>          after balancing and/or permuting. Permuting does not change
   *>          condition numbers (in exact arithmetic), but balancing does.
   *> \endverbatim
   *>
   *> \param[in] JOBVL
   *> \verbatim
   *>          JOBVL is CHARACTER*1
   *>          = 'N': left eigenvectors of A are not computed;
   *>          = 'V': left eigenvectors of A are computed.
   *>          If SENSE = 'E' or 'B', JOBVL must = 'V'.
   *> \endverbatim
   *>
   *> \param[in] JOBVR
   *> \verbatim
   *>          JOBVR is CHARACTER*1
   *>          = 'N': right eigenvectors of A are not computed;
   *>          = 'V': right eigenvectors of A are computed.
   *>          If SENSE = 'E' or 'B', JOBVR must = 'V'.
   *> \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 right eigenvectors only;
   *>          = 'B': Computed for eigenvalues and right eigenvectors.
   *>
   *>          If SENSE = 'E' or 'B', both left and right eigenvectors
   *>          must also be computed (JOBVL = 'V' and JOBVR = 'V').
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix A. N >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   *>          On entry, the N-by-N matrix A.
   *>          On exit, A has been overwritten.  If JOBVL = 'V' or
   *>          JOBVR = 'V', A contains the real Schur form of the balanced
   *>          version of the input matrix A.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] WR
   *> \verbatim
   *>          WR is DOUBLE PRECISION array, dimension (N)
   *> \endverbatim
   *>
   *> \param[out] WI
   *> \verbatim
   *>          WI is DOUBLE PRECISION array, dimension (N)
   *>          WR and WI contain the real and imaginary parts,
   *>          respectively, of the computed eigenvalues.  Complex
   *>          conjugate pairs of eigenvalues will appear consecutively
   *>          with the eigenvalue having the positive imaginary part
   *>          first.
   *> \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 JOBVL = 'N', VL is not referenced.
   *>          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)-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).
   *> \endverbatim
   *>
   *> \param[in] LDVL
   *> \verbatim
   *>          LDVL is INTEGER
   *>          The leading dimension of the array VL.  LDVL >= 1; 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 JOBVR = 'N', VR is not referenced.
   *>          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)-st 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).
   *> \endverbatim
   *>
   *> \param[in] LDVR
   *> \verbatim
   *>          LDVR is INTEGER
   *>          The leading dimension of the array 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 determined when A was
   *>          balanced.  The balanced A(i,j) = 0 if I > J and
   *>          J = 1,...,ILO-1 or I = IHI+1,...,N.
   *> \endverbatim
   *>
   *> \param[out] SCALE
   *> \verbatim
   *>          SCALE is DOUBLE PRECISION array, dimension (N)
   *>          Details of the permutations and scaling factors applied
   *>          when balancing A.  If P(j) is the index of the row and column
   *>          interchanged with row and column j, and D(j) is the scaling
   *>          factor applied to row and column j, then
   *>          SCALE(J) = P(J),    for J = 1,...,ILO-1
   *>                   = D(J),    for J = ILO,...,IHI
   *>                   = P(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 (the maximum
   *>          of the sum of absolute values of elements of any column).
   *> \endverbatim
   *>
   *> \param[out] RCONDE
   *> \verbatim
   *>          RCONDE is DOUBLE PRECISION array, dimension (N)
   *>          RCONDE(j) is the reciprocal condition number of the j-th
   *>          eigenvalue.
   *> \endverbatim
   *>
   *> \param[out] RCONDV
   *> \verbatim
   *>          RCONDV is DOUBLE PRECISION array, dimension (N)
   *>          RCONDV(j) is the reciprocal condition number of the j-th
   *>          right eigenvector.
   *> \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.   If SENSE = 'N' or 'E',
   *>          LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
   *>          LWORK >= 3*N.  If SENSE = 'V' or 'B', LWORK >= N*(N+6).
   *>          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] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (2*N-2)
   *>          If SENSE = 'N' or 'E', 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.
   *>          > 0:  if INFO = i, the QR algorithm failed to compute all the
   *>                eigenvalues, and no eigenvectors or condition numbers
   *>                have been computed; elements 1:ILO-1 and i+1:N of WR
   *>                and WI contain eigenvalues which have converged.
   *> \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 DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,        SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
      $                   VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,       $                   VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
      $                   RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )       $                   RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.3.1) --  *  -- 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..--
 *  -- April 2011                                                      --  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          BALANC, JOBVL, JOBVR, SENSE        CHARACTER          BALANC, JOBVL, JOBVR, SENSE
Line 19 Line 320
      $                   WI( * ), WORK( * ), WR( * )       $                   WI( * ), WORK( * ), WR( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DGEEVX computes for an N-by-N real nonsymmetric matrix A, the  
 *  eigenvalues and, optionally, the left and/or right eigenvectors.  
 *  
 *  Optionally also, it computes a balancing transformation to improve  
 *  the conditioning of the eigenvalues and eigenvectors (ILO, IHI,  
 *  SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues  
 *  (RCONDE), and reciprocal condition numbers for the right  
 *  eigenvectors (RCONDV).  
 *  
 *  The right eigenvector v(j) of A satisfies  
 *                   A * v(j) = lambda(j) * v(j)  
 *  where lambda(j) is its eigenvalue.  
 *  The left eigenvector u(j) of A satisfies  
 *                u(j)**T * A = lambda(j) * u(j)**T  
 *  where u(j)**T denotes the transpose of u(j).  
 *  
 *  The computed eigenvectors are normalized to have Euclidean norm  
 *  equal to 1 and largest component real.  
 *  
 *  Balancing a matrix means permuting the rows and columns to make it  
 *  more nearly upper triangular, and applying a diagonal similarity  
 *  transformation D * A * D**(-1), where D is a diagonal matrix, to  
 *  make its rows and columns closer in norm and the condition numbers  
 *  of its eigenvalues and eigenvectors smaller.  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 4.10.2 of the LAPACK  
 *  Users' Guide.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  BALANC  (input) CHARACTER*1  
 *          Indicates how the input matrix should be diagonally scaled  
 *          and/or permuted to improve the conditioning of its  
 *          eigenvalues.  
 *          = 'N': Do not diagonally scale or permute;  
 *          = 'P': Perform permutations to make the matrix more nearly  
 *                 upper triangular. Do not diagonally scale;  
 *          = 'S': Diagonally scale the matrix, i.e. replace A by  
 *                 D*A*D**(-1), where D is a diagonal matrix chosen  
 *                 to make the rows and columns of A more equal in  
 *                 norm. Do not permute;  
 *          = 'B': Both diagonally scale and permute A.  
 *  
 *          Computed reciprocal condition numbers will be for the matrix  
 *          after balancing and/or permuting. Permuting does not change  
 *          condition numbers (in exact arithmetic), but balancing does.  
 *  
 *  JOBVL   (input) CHARACTER*1  
 *          = 'N': left eigenvectors of A are not computed;  
 *          = 'V': left eigenvectors of A are computed.  
 *          If SENSE = 'E' or 'B', JOBVL must = 'V'.  
 *  
 *  JOBVR   (input) CHARACTER*1  
 *          = 'N': right eigenvectors of A are not computed;  
 *          = 'V': right eigenvectors of A are computed.  
 *          If SENSE = 'E' or 'B', JOBVR must = 'V'.  
 *  
 *  SENSE   (input) CHARACTER*1  
 *          Determines which reciprocal condition numbers are computed.  
 *          = 'N': None are computed;  
 *          = 'E': Computed for eigenvalues only;  
 *          = 'V': Computed for right eigenvectors only;  
 *          = 'B': Computed for eigenvalues and right eigenvectors.  
 *  
 *          If SENSE = 'E' or 'B', both left and right eigenvectors  
 *          must also be computed (JOBVL = 'V' and JOBVR = 'V').  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrix A. N >= 0.  
 *  
 *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)  
 *          On entry, the N-by-N matrix A.  
 *          On exit, A has been overwritten.  If JOBVL = 'V' or  
 *          JOBVR = 'V', A contains the real Schur form of the balanced  
 *          version of the input matrix A.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,N).  
 *  
 *  WR      (output) DOUBLE PRECISION array, dimension (N)  
 *  WI      (output) DOUBLE PRECISION array, dimension (N)  
 *          WR and WI contain the real and imaginary parts,  
 *          respectively, of the computed eigenvalues.  Complex  
 *          conjugate pairs of eigenvalues will appear consecutively  
 *          with the eigenvalue having the positive imaginary part  
 *          first.  
 *  
 *  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 JOBVL = 'N', VL is not referenced.  
 *          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)-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).  
 *  
 *  LDVL    (input) INTEGER  
 *          The leading dimension of the array VL.  LDVL >= 1; 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 JOBVR = 'N', VR is not referenced.  
 *          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)-st 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).  
 *  
 *  LDVR    (input) INTEGER  
 *          The leading dimension of the array VR.  LDVR >= 1, and if  
 *          JOBVR = 'V', LDVR >= N.  
 *  
 *  ILO     (output) INTEGER  
 *  IHI     (output) INTEGER  
 *          ILO and IHI are integer values determined when A was  
 *          balanced.  The balanced A(i,j) = 0 if I > J and  
 *          J = 1,...,ILO-1 or I = IHI+1,...,N.  
 *  
 *  SCALE   (output) DOUBLE PRECISION array, dimension (N)  
 *          Details of the permutations and scaling factors applied  
 *          when balancing A.  If P(j) is the index of the row and column  
 *          interchanged with row and column j, and D(j) is the scaling  
 *          factor applied to row and column j, then  
 *          SCALE(J) = P(J),    for J = 1,...,ILO-1  
 *                   = D(J),    for J = ILO,...,IHI  
 *                   = P(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 (the maximum  
 *          of the sum of absolute values of elements of any column).  
 *  
 *  RCONDE  (output) DOUBLE PRECISION array, dimension (N)  
 *          RCONDE(j) is the reciprocal condition number of the j-th  
 *          eigenvalue.  
 *  
 *  RCONDV  (output) DOUBLE PRECISION array, dimension (N)  
 *          RCONDV(j) is the reciprocal condition number of the j-th  
 *          right eigenvector.  
 *  
 *  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.   If SENSE = 'N' or 'E',  
 *          LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',  
 *          LWORK >= 3*N.  If SENSE = 'V' or 'B', LWORK >= N*(N+6).  
 *          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.  
 *  
 *  IWORK   (workspace) INTEGER array, dimension (2*N-2)  
 *          If SENSE = 'N' or 'E', not referenced.  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value.  
 *          > 0:  if INFO = i, the QR algorithm failed to compute all the  
 *                eigenvalues, and no eigenvectors or condition numbers  
 *                have been computed; elements 1:ILO-1 and i+1:N of WR  
 *                and WI contain eigenvalues which have converged.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.8  
changed lines
  Added in v.1.9

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