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version 1.4, 2010/08/06 15:32:23 version 1.17, 2017/06/17 10:53:47
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   *> \brief <b> DGEEV 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 DGEEV + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeev.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeev.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeev.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,
   *                         LDVR, WORK, LWORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBVL, JOBVR
   *       INTEGER            INFO, LDA, LDVL, LDVR, LWORK, N
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
   *      $                   WI( * ), WORK( * ), WR( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DGEEV computes for an N-by-N real nonsymmetric matrix A, the
   *> eigenvalues and, optionally, the left and/or right eigenvectors.
   *>
   *> 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)**H * A = lambda(j) * u(j)**H
   *> where u(j)**H denotes the conjugate-transpose of u(j).
   *>
   *> The computed eigenvectors are normalized to have Euclidean norm
   *> equal to 1 and largest component real.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOBVL
   *> \verbatim
   *>          JOBVL is CHARACTER*1
   *>          = 'N': left eigenvectors of A are not computed;
   *>          = 'V': left eigenvectors of A are computed.
   *> \endverbatim
   *>
   *> \param[in] JOBVR
   *> \verbatim
   *>          JOBVR is CHARACTER*1
   *>          = 'N': right eigenvectors of A are not computed;
   *>          = 'V': right eigenvectors of A are computed.
   *> \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.
   *> \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 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; 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(1,3*N), and
   *>          if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N.  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] 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 have been computed;
   *>                elements 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 June 2016
   *
   *  @precisions fortran d -> s
   *
   *> \ingroup doubleGEeigen
   *
   *  =====================================================================
       SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,        SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,
      $                  LDVR, WORK, LWORK, INFO )       $                  LDVR, WORK, LWORK, INFO )
         implicit none
 *  *
 *  -- 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  *     June 2016
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOBVL, JOBVR        CHARACTER          JOBVL, JOBVR
Line 15 Line 206
      $                   WI( * ), WORK( * ), WR( * )       $                   WI( * ), WORK( * ), WR( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DGEEV computes for an N-by-N real nonsymmetric matrix A, the  
 *  eigenvalues and, optionally, the left and/or right eigenvectors.  
 *  
 *  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)**H * A = lambda(j) * u(j)**H  
 *  where u(j)**H denotes the conjugate transpose of u(j).  
 *  
 *  The computed eigenvectors are normalized to have Euclidean norm  
 *  equal to 1 and largest component real.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  JOBVL   (input) CHARACTER*1  
 *          = 'N': left eigenvectors of A are not computed;  
 *          = 'V': left eigenvectors of A are computed.  
 *  
 *  JOBVR   (input) CHARACTER*1  
 *          = 'N': right eigenvectors of A are not computed;  
 *          = 'V': right eigenvectors of A are computed.  
 *  
 *  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.  
 *  
 *  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 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; 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(1,3*N), and  
 *          if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N.  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.  
 *  
 *  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 have been computed;  
 *                elements i+1:N of WR and WI contain eigenvalues which  
 *                have converged.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Line 121 Line 216
       LOGICAL            LQUERY, SCALEA, WANTVL, WANTVR        LOGICAL            LQUERY, SCALEA, WANTVL, WANTVR
       CHARACTER          SIDE        CHARACTER          SIDE
       INTEGER            HSWORK, I, IBAL, IERR, IHI, ILO, ITAU, IWRK, K,        INTEGER            HSWORK, I, IBAL, IERR, IHI, ILO, ITAU, IWRK, K,
      $                   MAXWRK, MINWRK, NOUT       $                   LWORK_TREVC, MAXWRK, MINWRK, NOUT
       DOUBLE PRECISION   ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,        DOUBLE PRECISION   ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
      $                   SN       $                   SN
 *     ..  *     ..
Line 131 Line 226
 *     ..  *     ..
 *     .. External Subroutines ..  *     .. External Subroutines ..
       EXTERNAL           DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLABAD, DLACPY,        EXTERNAL           DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLABAD, DLACPY,
      $                   DLARTG, DLASCL, DORGHR, DROT, DSCAL, DTREVC,       $                   DLARTG, DLASCL, DORGHR, DROT, DSCAL, DTREVC3,
      $                   XERBLA       $                   XERBLA
 *     ..  *     ..
 *     .. External Functions ..  *     .. External Functions ..
Line 187 Line 282
                MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,                 MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
      $                       'DORGHR', ' ', N, 1, N, -1 ) )       $                       'DORGHR', ' ', N, 1, N, -1 ) )
                CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL,                 CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL,
      $                WORK, -1, INFO )       $                      WORK, -1, INFO )
                HSWORK = WORK( 1 )                 HSWORK = INT( WORK(1) )
                MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )                 MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
                  CALL DTREVC3( 'L', 'B', SELECT, N, A, LDA,
        $                       VL, LDVL, VR, LDVR, N, NOUT,
        $                       WORK, -1, IERR )
                  LWORK_TREVC = INT( WORK(1) )
                  MAXWRK = MAX( MAXWRK, N + LWORK_TREVC )
                MAXWRK = MAX( MAXWRK, 4*N )                 MAXWRK = MAX( MAXWRK, 4*N )
             ELSE IF( WANTVR ) THEN              ELSE IF( WANTVR ) THEN
                MINWRK = 4*N                 MINWRK = 4*N
                MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,                 MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
      $                       'DORGHR', ' ', N, 1, N, -1 ) )       $                       'DORGHR', ' ', N, 1, N, -1 ) )
                CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR,                 CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR,
      $                WORK, -1, INFO )       $                      WORK, -1, INFO )
                HSWORK = WORK( 1 )                 HSWORK = INT( WORK(1) )
                MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )                 MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
                  CALL DTREVC3( 'R', 'B', SELECT, N, A, LDA,
        $                       VL, LDVL, VR, LDVR, N, NOUT,
        $                       WORK, -1, IERR )
                  LWORK_TREVC = INT( WORK(1) )
                  MAXWRK = MAX( MAXWRK, N + LWORK_TREVC )
                MAXWRK = MAX( MAXWRK, 4*N )                 MAXWRK = MAX( MAXWRK, 4*N )
             ELSE               ELSE
                MINWRK = 3*N                 MINWRK = 3*N
                CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR, LDVR,                 CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR, LDVR,
      $                WORK, -1, INFO )       $                      WORK, -1, INFO )
                HSWORK = WORK( 1 )                 HSWORK = INT( WORK(1) )
                MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )                 MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
             END IF              END IF
             MAXWRK = MAX( MAXWRK, MINWRK )              MAXWRK = MAX( MAXWRK, MINWRK )
Line 326 Line 431
      $                WORK( IWRK ), LWORK-IWRK+1, INFO )       $                WORK( IWRK ), LWORK-IWRK+1, INFO )
       END IF        END IF
 *  *
 *     If INFO > 0 from DHSEQR, then quit  *     If INFO .NE. 0 from DHSEQR, then quit
 *  *
       IF( INFO.GT.0 )        IF( INFO.NE.0 )
      $   GO TO 50       $   GO TO 50
 *  *
       IF( WANTVL .OR. WANTVR ) THEN        IF( WANTVL .OR. WANTVR ) THEN
 *  *
 *        Compute left and/or right eigenvectors  *        Compute left and/or right eigenvectors
 *        (Workspace: need 4*N)  *        (Workspace: need 4*N, prefer N + N + 2*N*NB)
 *  *
          CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,           CALL DTREVC3( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
      $                N, NOUT, WORK( IWRK ), IERR )       $                 N, NOUT, WORK( IWRK ), LWORK-IWRK+1, IERR )
       END IF        END IF
 *  *
       IF( WANTVL ) THEN        IF( WANTVL ) THEN
Removed from v.1.4  
changed lines
  Added in v.1.17

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