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version 1.8, 2011/07/22 07:38:13 version 1.9, 2011/11/21 20:43:06
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   *> \brief \b DTREVC
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DTREVC + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrevc.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrevc.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrevc.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
   *                          LDVR, MM, M, WORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          HOWMNY, SIDE
   *       INTEGER            INFO, LDT, LDVL, LDVR, M, MM, N
   *       ..
   *       .. Array Arguments ..
   *       LOGICAL            SELECT( * )
   *       DOUBLE PRECISION   T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
   *      $                   WORK( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DTREVC computes some or all of the right and/or left eigenvectors of
   *> a real upper quasi-triangular matrix T.
   *> Matrices of this type are produced by the Schur factorization of
   *> a real general matrix:  A = Q*T*Q**T, as computed by DHSEQR.
   *> 
   *> The right eigenvector x and the left eigenvector y of T corresponding
   *> to an eigenvalue w are defined by:
   *> 
   *>    T*x = w*x,     (y**T)*T = w*(y**T)
   *> 
   *> where y**T denotes the transpose of y.
   *> The eigenvalues are not input to this routine, but are read directly
   *> from the diagonal blocks of T.
   *> 
   *> This routine returns the matrices X and/or Y of right and left
   *> eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
   *> input matrix.  If Q is the orthogonal factor that reduces a matrix
   *> A to Schur form T, then Q*X and Q*Y are the matrices of right and
   *> left eigenvectors of A.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] SIDE
   *> \verbatim
   *>          SIDE is CHARACTER*1
   *>          = 'R':  compute right eigenvectors only;
   *>          = 'L':  compute left eigenvectors only;
   *>          = 'B':  compute both right and left eigenvectors.
   *> \endverbatim
   *>
   *> \param[in] HOWMNY
   *> \verbatim
   *>          HOWMNY is CHARACTER*1
   *>          = 'A':  compute all right and/or left eigenvectors;
   *>          = 'B':  compute all right and/or left eigenvectors,
   *>                  backtransformed by the matrices in VR and/or VL;
   *>          = 'S':  compute selected right and/or left eigenvectors,
   *>                  as indicated by the logical array SELECT.
   *> \endverbatim
   *>
   *> \param[in,out] SELECT
   *> \verbatim
   *>          SELECT is LOGICAL array, dimension (N)
   *>          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
   *>          computed.
   *>          If w(j) is a real eigenvalue, the corresponding real
   *>          eigenvector is computed if SELECT(j) is .TRUE..
   *>          If w(j) and w(j+1) are the real and imaginary parts of a
   *>          complex eigenvalue, the corresponding complex eigenvector is
   *>          computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
   *>          on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
   *>          .FALSE..
   *>          Not referenced if HOWMNY = 'A' or 'B'.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix T. N >= 0.
   *> \endverbatim
   *>
   *> \param[in] T
   *> \verbatim
   *>          T is DOUBLE PRECISION array, dimension (LDT,N)
   *>          The upper quasi-triangular matrix T in Schur canonical form.
   *> \endverbatim
   *>
   *> \param[in] LDT
   *> \verbatim
   *>          LDT is INTEGER
   *>          The leading dimension of the array T. LDT >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in,out] VL
   *> \verbatim
   *>          VL is DOUBLE PRECISION array, dimension (LDVL,MM)
   *>          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
   *>          contain an N-by-N matrix Q (usually the orthogonal matrix Q
   *>          of Schur vectors returned by DHSEQR).
   *>          On exit, if SIDE = 'L' or 'B', VL contains:
   *>          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
   *>          if HOWMNY = 'B', the matrix Q*Y;
   *>          if HOWMNY = 'S', the left eigenvectors of T specified by
   *>                           SELECT, stored consecutively in the columns
   *>                           of VL, in the same order as their
   *>                           eigenvalues.
   *>          A complex eigenvector corresponding to a complex eigenvalue
   *>          is stored in two consecutive columns, the first holding the
   *>          real part, and the second the imaginary part.
   *>          Not referenced if SIDE = 'R'.
   *> \endverbatim
   *>
   *> \param[in] LDVL
   *> \verbatim
   *>          LDVL is INTEGER
   *>          The leading dimension of the array VL.  LDVL >= 1, and if
   *>          SIDE = 'L' or 'B', LDVL >= N.
   *> \endverbatim
   *>
   *> \param[in,out] VR
   *> \verbatim
   *>          VR is DOUBLE PRECISION array, dimension (LDVR,MM)
   *>          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
   *>          contain an N-by-N matrix Q (usually the orthogonal matrix Q
   *>          of Schur vectors returned by DHSEQR).
   *>          On exit, if SIDE = 'R' or 'B', VR contains:
   *>          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
   *>          if HOWMNY = 'B', the matrix Q*X;
   *>          if HOWMNY = 'S', the right eigenvectors of T specified by
   *>                           SELECT, stored consecutively in the columns
   *>                           of VR, in the same order as their
   *>                           eigenvalues.
   *>          A complex eigenvector corresponding to a complex eigenvalue
   *>          is stored in two consecutive columns, the first holding the
   *>          real part and the second the imaginary part.
   *>          Not referenced if SIDE = 'L'.
   *> \endverbatim
   *>
   *> \param[in] LDVR
   *> \verbatim
   *>          LDVR is INTEGER
   *>          The leading dimension of the array VR.  LDVR >= 1, and if
   *>          SIDE = 'R' or 'B', LDVR >= N.
   *> \endverbatim
   *>
   *> \param[in] MM
   *> \verbatim
   *>          MM is INTEGER
   *>          The number of columns in the arrays VL and/or VR. MM >= M.
   *> \endverbatim
   *>
   *> \param[out] M
   *> \verbatim
   *>          M is INTEGER
   *>          The number of columns in the arrays VL and/or VR actually
   *>          used to store the eigenvectors.
   *>          If HOWMNY = 'A' or 'B', M is set to N.
   *>          Each selected real eigenvector occupies one column and each
   *>          selected complex eigenvector occupies two columns.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (3*N)
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0:  successful exit
   *>          < 0:  if INFO = -i, the i-th argument had an illegal value
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleOTHERcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  The algorithm used in this program is basically backward (forward)
   *>  substitution, with scaling to make the the code robust against
   *>  possible overflow.
   *>
   *>  Each eigenvector is normalized so that the element of largest
   *>  magnitude has magnitude 1; here the magnitude of a complex number
   *>  (x,y) is taken to be |x| + |y|.
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,        SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
      $                   LDVR, MM, M, WORK, INFO )       $                   LDVR, MM, M, WORK, INFO )
 *  *
 *  -- LAPACK routine (version 3.3.1) --  *  -- LAPACK computational 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          HOWMNY, SIDE        CHARACTER          HOWMNY, SIDE
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      $                   WORK( * )       $                   WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DTREVC computes some or all of the right and/or left eigenvectors of  
 *  a real upper quasi-triangular matrix T.  
 *  Matrices of this type are produced by the Schur factorization of  
 *  a real general matrix:  A = Q*T*Q**T, as computed by DHSEQR.  
 *    
 *  The right eigenvector x and the left eigenvector y of T corresponding  
 *  to an eigenvalue w are defined by:  
 *    
 *     T*x = w*x,     (y**T)*T = w*(y**T)  
 *    
 *  where y**T denotes the transpose of y.  
 *  The eigenvalues are not input to this routine, but are read directly  
 *  from the diagonal blocks of T.  
 *    
 *  This routine returns the matrices X and/or Y of right and left  
 *  eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an  
 *  input matrix.  If Q is the orthogonal factor that reduces a matrix  
 *  A to Schur form T, then Q*X and Q*Y are the matrices of right and  
 *  left eigenvectors of A.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  SIDE    (input) CHARACTER*1  
 *          = 'R':  compute right eigenvectors only;  
 *          = 'L':  compute left eigenvectors only;  
 *          = 'B':  compute both right and left eigenvectors.  
 *  
 *  HOWMNY  (input) CHARACTER*1  
 *          = 'A':  compute all right and/or left eigenvectors;  
 *          = 'B':  compute all right and/or left eigenvectors,  
 *                  backtransformed by the matrices in VR and/or VL;  
 *          = 'S':  compute selected right and/or left eigenvectors,  
 *                  as indicated by the logical array SELECT.  
 *  
 *  SELECT  (input/output) LOGICAL array, dimension (N)  
 *          If HOWMNY = 'S', SELECT specifies the eigenvectors to be  
 *          computed.  
 *          If w(j) is a real eigenvalue, the corresponding real  
 *          eigenvector is computed if SELECT(j) is .TRUE..  
 *          If w(j) and w(j+1) are the real and imaginary parts of a  
 *          complex eigenvalue, the corresponding complex eigenvector is  
 *          computed if either SELECT(j) or SELECT(j+1) is .TRUE., and  
 *          on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to  
 *          .FALSE..  
 *          Not referenced if HOWMNY = 'A' or 'B'.  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrix T. N >= 0.  
 *  
 *  T       (input) DOUBLE PRECISION array, dimension (LDT,N)  
 *          The upper quasi-triangular matrix T in Schur canonical form.  
 *  
 *  LDT     (input) INTEGER  
 *          The leading dimension of the array T. LDT >= max(1,N).  
 *  
 *  VL      (input/output) DOUBLE PRECISION array, dimension (LDVL,MM)  
 *          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must  
 *          contain an N-by-N matrix Q (usually the orthogonal matrix Q  
 *          of Schur vectors returned by DHSEQR).  
 *          On exit, if SIDE = 'L' or 'B', VL contains:  
 *          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;  
 *          if HOWMNY = 'B', the matrix Q*Y;  
 *          if HOWMNY = 'S', the left eigenvectors of T specified by  
 *                           SELECT, stored consecutively in the columns  
 *                           of VL, in the same order as their  
 *                           eigenvalues.  
 *          A complex eigenvector corresponding to a complex eigenvalue  
 *          is stored in two consecutive columns, the first holding the  
 *          real part, and the second the imaginary part.  
 *          Not referenced if SIDE = 'R'.  
 *  
 *  LDVL    (input) INTEGER  
 *          The leading dimension of the array VL.  LDVL >= 1, and if  
 *          SIDE = 'L' or 'B', LDVL >= N.  
 *  
 *  VR      (input/output) DOUBLE PRECISION array, dimension (LDVR,MM)  
 *          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must  
 *          contain an N-by-N matrix Q (usually the orthogonal matrix Q  
 *          of Schur vectors returned by DHSEQR).  
 *          On exit, if SIDE = 'R' or 'B', VR contains:  
 *          if HOWMNY = 'A', the matrix X of right eigenvectors of T;  
 *          if HOWMNY = 'B', the matrix Q*X;  
 *          if HOWMNY = 'S', the right eigenvectors of T specified by  
 *                           SELECT, stored consecutively in the columns  
 *                           of VR, in the same order as their  
 *                           eigenvalues.  
 *          A complex eigenvector corresponding to a complex eigenvalue  
 *          is stored in two consecutive columns, the first holding the  
 *          real part and the second the imaginary part.  
 *          Not referenced if SIDE = 'L'.  
 *  
 *  LDVR    (input) INTEGER  
 *          The leading dimension of the array VR.  LDVR >= 1, and if  
 *          SIDE = 'R' or 'B', LDVR >= N.  
 *  
 *  MM      (input) INTEGER  
 *          The number of columns in the arrays VL and/or VR. MM >= M.  
 *  
 *  M       (output) INTEGER  
 *          The number of columns in the arrays VL and/or VR actually  
 *          used to store the eigenvectors.  
 *          If HOWMNY = 'A' or 'B', M is set to N.  
 *          Each selected real eigenvector occupies one column and each  
 *          selected complex eigenvector occupies two columns.  
 *  
 *  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  The algorithm used in this program is basically backward (forward)  
 *  substitution, with scaling to make the the code robust against  
 *  possible overflow.  
 *  
 *  Each eigenvector is normalized so that the element of largest  
 *  magnitude has magnitude 1; here the magnitude of a complex number  
 *  (x,y) is taken to be |x| + |y|.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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  Added in v.1.9

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