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version 1.6, 2010/08/13 21:03:58 version 1.19, 2023/08/07 08:39:08
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   *> \brief \b DSYGVD
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download DSYGVD + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygvd.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygvd.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygvd.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DSYGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
   *                          LWORK, IWORK, LIWORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBZ, UPLO
   *       INTEGER            INFO, ITYPE, LDA, LDB, LIWORK, LWORK, N
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IWORK( * )
   *       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DSYGVD computes all the eigenvalues, and optionally, the eigenvectors
   *> of a real generalized symmetric-definite eigenproblem, of the form
   *> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
   *> B are assumed to be symmetric and B is also positive definite.
   *> If eigenvectors are desired, it uses a divide and conquer algorithm.
   *>
   *> The divide and conquer algorithm makes very mild assumptions about
   *> floating point arithmetic. It will work on machines with a guard
   *> digit in add/subtract, or on those binary machines without guard
   *> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
   *> Cray-2. It could conceivably fail on hexadecimal or decimal machines
   *> without guard digits, but we know of none.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] ITYPE
   *> \verbatim
   *>          ITYPE is INTEGER
   *>          Specifies the problem type to be solved:
   *>          = 1:  A*x = (lambda)*B*x
   *>          = 2:  A*B*x = (lambda)*x
   *>          = 3:  B*A*x = (lambda)*x
   *> \endverbatim
   *>
   *> \param[in] JOBZ
   *> \verbatim
   *>          JOBZ is CHARACTER*1
   *>          = 'N':  Compute eigenvalues only;
   *>          = 'V':  Compute eigenvalues and eigenvectors.
   *> \endverbatim
   *>
   *> \param[in] UPLO
   *> \verbatim
   *>          UPLO is CHARACTER*1
   *>          = 'U':  Upper triangles of A and B are stored;
   *>          = 'L':  Lower triangles of A and B are stored.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrices A and B.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA, N)
   *>          On entry, the symmetric matrix A.  If UPLO = 'U', the
   *>          leading N-by-N upper triangular part of A contains the
   *>          upper triangular part of the matrix A.  If UPLO = 'L',
   *>          the leading N-by-N lower triangular part of A contains
   *>          the lower triangular part of the matrix A.
   *>
   *>          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
   *>          matrix Z of eigenvectors.  The eigenvectors are normalized
   *>          as follows:
   *>          if ITYPE = 1 or 2, Z**T*B*Z = I;
   *>          if ITYPE = 3, Z**T*inv(B)*Z = I.
   *>          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
   *>          or the lower triangle (if UPLO='L') of A, including the
   *>          diagonal, is destroyed.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in,out] B
   *> \verbatim
   *>          B is DOUBLE PRECISION array, dimension (LDB, N)
   *>          On entry, the symmetric matrix B.  If UPLO = 'U', the
   *>          leading N-by-N upper triangular part of B contains the
   *>          upper triangular part of the matrix B.  If UPLO = 'L',
   *>          the leading N-by-N lower triangular part of B contains
   *>          the lower triangular part of the matrix B.
   *>
   *>          On exit, if INFO <= N, the part of B containing the matrix is
   *>          overwritten by the triangular factor U or L from the Cholesky
   *>          factorization B = U**T*U or B = L*L**T.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array B.  LDB >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] W
   *> \verbatim
   *>          W is DOUBLE PRECISION array, dimension (N)
   *>          If INFO = 0, the eigenvalues in ascending order.
   *> \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 N <= 1,               LWORK >= 1.
   *>          If JOBZ = 'N' and N > 1, LWORK >= 2*N+1.
   *>          If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
   *>
   *>          If LWORK = -1, then a workspace query is assumed; the routine
   *>          only calculates the optimal sizes of the WORK and IWORK
   *>          arrays, returns these values as the first entries of the WORK
   *>          and IWORK arrays, and no error message related to LWORK or
   *>          LIWORK is issued by XERBLA.
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
   *>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
   *> \endverbatim
   *>
   *> \param[in] LIWORK
   *> \verbatim
   *>          LIWORK is INTEGER
   *>          The dimension of the array IWORK.
   *>          If N <= 1,                LIWORK >= 1.
   *>          If JOBZ  = 'N' and N > 1, LIWORK >= 1.
   *>          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
   *>
   *>          If LIWORK = -1, then a workspace query is assumed; the
   *>          routine only calculates the optimal sizes of the WORK and
   *>          IWORK arrays, returns these values as the first entries of
   *>          the WORK and IWORK arrays, and no error message related to
   *>          LWORK or LIWORK 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:  DPOTRF or DSYEVD returned an error code:
   *>             <= N:  if INFO = i and JOBZ = 'N', then the algorithm
   *>                    failed to converge; i off-diagonal elements of an
   *>                    intermediate tridiagonal form did not converge to
   *>                    zero;
   *>                    if INFO = i and JOBZ = 'V', then the algorithm
   *>                    failed to compute an eigenvalue while working on
   *>                    the submatrix lying in rows and columns INFO/(N+1)
   *>                    through mod(INFO,N+1);
   *>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
   *>                    minor of order i of B is not positive definite.
   *>                    The factorization of B could not be completed and
   *>                    no eigenvalues or eigenvectors were computed.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \ingroup doubleSYeigen
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  Modified so that no backsubstitution is performed if DSYEVD fails to
   *>  converge (NEIG in old code could be greater than N causing out of
   *>  bounds reference to A - reported by Ralf Meyer).  Also corrected the
   *>  description of INFO and the test on ITYPE. Sven, 16 Feb 05.
   *> \endverbatim
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
   *>
   *  =====================================================================
       SUBROUTINE DSYGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,        SUBROUTINE DSYGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
      $                   LWORK, IWORK, LIWORK, INFO )       $                   LWORK, IWORK, LIWORK, 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          JOBZ, UPLO        CHARACTER          JOBZ, UPLO
Line 15 Line 238
       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), W( * ), WORK( * )        DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DSYGVD computes all the eigenvalues, and optionally, the eigenvectors  
 *  of a real generalized symmetric-definite eigenproblem, of the form  
 *  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and  
 *  B are assumed to be symmetric and B is also positive definite.  
 *  If eigenvectors are desired, it uses a divide and conquer algorithm.  
 *  
 *  The divide and conquer algorithm makes very mild assumptions about  
 *  floating point arithmetic. It will work on machines with a guard  
 *  digit in add/subtract, or on those binary machines without guard  
 *  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or  
 *  Cray-2. It could conceivably fail on hexadecimal or decimal machines  
 *  without guard digits, but we know of none.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  ITYPE   (input) INTEGER  
 *          Specifies the problem type to be solved:  
 *          = 1:  A*x = (lambda)*B*x  
 *          = 2:  A*B*x = (lambda)*x  
 *          = 3:  B*A*x = (lambda)*x  
 *  
 *  JOBZ    (input) CHARACTER*1  
 *          = 'N':  Compute eigenvalues only;  
 *          = 'V':  Compute eigenvalues and eigenvectors.  
 *  
 *  UPLO    (input) CHARACTER*1  
 *          = 'U':  Upper triangles of A and B are stored;  
 *          = 'L':  Lower triangles of A and B are stored.  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrices A and B.  N >= 0.  
 *  
 *  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)  
 *          On entry, the symmetric matrix A.  If UPLO = 'U', the  
 *          leading N-by-N upper triangular part of A contains the  
 *          upper triangular part of the matrix A.  If UPLO = 'L',  
 *          the leading N-by-N lower triangular part of A contains  
 *          the lower triangular part of the matrix A.  
 *  
 *          On exit, if JOBZ = 'V', then if INFO = 0, A contains the  
 *          matrix Z of eigenvectors.  The eigenvectors are normalized  
 *          as follows:  
 *          if ITYPE = 1 or 2, Z**T*B*Z = I;  
 *          if ITYPE = 3, Z**T*inv(B)*Z = I.  
 *          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')  
 *          or the lower triangle (if UPLO='L') of A, including the  
 *          diagonal, is destroyed.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,N).  
 *  
 *  B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)  
 *          On entry, the symmetric matrix B.  If UPLO = 'U', the  
 *          leading N-by-N upper triangular part of B contains the  
 *          upper triangular part of the matrix B.  If UPLO = 'L',  
 *          the leading N-by-N lower triangular part of B contains  
 *          the lower triangular part of the matrix B.  
 *  
 *          On exit, if INFO <= N, the part of B containing the matrix is  
 *          overwritten by the triangular factor U or L from the Cholesky  
 *          factorization B = U**T*U or B = L*L**T.  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of the array B.  LDB >= max(1,N).  
 *  
 *  W       (output) DOUBLE PRECISION array, dimension (N)  
 *          If INFO = 0, the eigenvalues in ascending order.  
 *  
 *  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 N <= 1,               LWORK >= 1.  
 *          If JOBZ = 'N' and N > 1, LWORK >= 2*N+1.  
 *          If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.  
 *  
 *          If LWORK = -1, then a workspace query is assumed; the routine  
 *          only calculates the optimal sizes of the WORK and IWORK  
 *          arrays, returns these values as the first entries of the WORK  
 *          and IWORK arrays, and no error message related to LWORK or  
 *          LIWORK is issued by XERBLA.  
 *  
 *  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))  
 *          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.  
 *  
 *  LIWORK  (input) INTEGER  
 *          The dimension of the array IWORK.  
 *          If N <= 1,                LIWORK >= 1.  
 *          If JOBZ  = 'N' and N > 1, LIWORK >= 1.  
 *          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.  
 *  
 *          If LIWORK = -1, then a workspace query is assumed; the  
 *          routine only calculates the optimal sizes of the WORK and  
 *          IWORK arrays, returns these values as the first entries of  
 *          the WORK and IWORK arrays, and no error message related to  
 *          LWORK or LIWORK is issued by XERBLA.  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value  
 *          > 0:  DPOTRF or DSYEVD returned an error code:  
 *             <= N:  if INFO = i and JOBZ = 'N', then the algorithm  
 *                    failed to converge; i off-diagonal elements of an  
 *                    intermediate tridiagonal form did not converge to  
 *                    zero;  
 *                    if INFO = i and JOBZ = 'V', then the algorithm  
 *                    failed to compute an eigenvalue while working on  
 *                    the submatrix lying in rows and columns INFO/(N+1)  
 *                    through mod(INFO,N+1);  
 *             > N:   if INFO = N + i, for 1 <= i <= N, then the leading  
 *                    minor of order i of B is not positive definite.  
 *                    The factorization of B could not be completed and  
 *                    no eigenvalues or eigenvectors were computed.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  Based on contributions by  
 *     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA  
 *  
 *  Modified so that no backsubstitution is performed if DSYEVD fails to  
 *  converge (NEIG in old code could be greater than N causing out of  
 *  bounds reference to A - reported by Ralf Meyer).  Also corrected the  
 *  description of INFO and the test on ITYPE. Sven, 16 Feb 05.  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Line 236 Line 330
       CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )        CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
       CALL DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK, LIWORK,        CALL DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK, LIWORK,
      $             INFO )       $             INFO )
       LOPT = MAX( DBLE( LOPT ), DBLE( WORK( 1 ) ) )        LOPT = INT( MAX( DBLE( LOPT ), DBLE( WORK( 1 ) ) ) )
       LIOPT = MAX( DBLE( LIOPT ), DBLE( IWORK( 1 ) ) )        LIOPT = INT( MAX( DBLE( LIOPT ), DBLE( IWORK( 1 ) ) ) )
 *  *
       IF( WANTZ .AND. INFO.EQ.0 ) THEN        IF( WANTZ .AND. INFO.EQ.0 ) THEN
 *  *
Line 246 Line 340
          IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN           IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
 *  *
 *           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;  *           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
 *           backtransform eigenvectors: x = inv(L)'*y or inv(U)*y  *           backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
 *  *
             IF( UPPER ) THEN              IF( UPPER ) THEN
                TRANS = 'N'                 TRANS = 'N'
Line 260 Line 354
          ELSE IF( ITYPE.EQ.3 ) THEN           ELSE IF( ITYPE.EQ.3 ) THEN
 *  *
 *           For B*A*x=(lambda)*x;  *           For B*A*x=(lambda)*x;
 *           backtransform eigenvectors: x = L*y or U'*y  *           backtransform eigenvectors: x = L*y or U**T*y
 *  *
             IF( UPPER ) THEN              IF( UPPER ) THEN
                TRANS = 'T'                 TRANS = 'T'
Removed from v.1.6  
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  Added in v.1.19

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