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version 1.6, 2010/08/13 21:03:57 version 1.19, 2023/08/07 08:39:05
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   *> \brief \b DSBGVD
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download DSBGVD + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbgvd.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbgvd.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbgvd.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W,
   *                          Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBZ, UPLO
   *       INTEGER            INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IWORK( * )
   *       DOUBLE PRECISION   AB( LDAB, * ), BB( LDBB, * ), W( * ),
   *      $                   WORK( * ), Z( LDZ, * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DSBGVD computes all the eigenvalues, and optionally, the eigenvectors
   *> of a real generalized symmetric-definite banded eigenproblem, of the
   *> form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric and
   *> banded, 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] 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] KA
   *> \verbatim
   *>          KA is INTEGER
   *>          The number of superdiagonals of the matrix A if UPLO = 'U',
   *>          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.
   *> \endverbatim
   *>
   *> \param[in] KB
   *> \verbatim
   *>          KB is INTEGER
   *>          The number of superdiagonals of the matrix B if UPLO = 'U',
   *>          or the number of subdiagonals if UPLO = 'L'.  KB >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] AB
   *> \verbatim
   *>          AB is DOUBLE PRECISION array, dimension (LDAB, N)
   *>          On entry, the upper or lower triangle of the symmetric band
   *>          matrix A, stored in the first ka+1 rows of the array.  The
   *>          j-th column of A is stored in the j-th column of the array AB
   *>          as follows:
   *>          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
   *>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
   *>
   *>          On exit, the contents of AB are destroyed.
   *> \endverbatim
   *>
   *> \param[in] LDAB
   *> \verbatim
   *>          LDAB is INTEGER
   *>          The leading dimension of the array AB.  LDAB >= KA+1.
   *> \endverbatim
   *>
   *> \param[in,out] BB
   *> \verbatim
   *>          BB is DOUBLE PRECISION array, dimension (LDBB, N)
   *>          On entry, the upper or lower triangle of the symmetric band
   *>          matrix B, stored in the first kb+1 rows of the array.  The
   *>          j-th column of B is stored in the j-th column of the array BB
   *>          as follows:
   *>          if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
   *>          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
   *>
   *>          On exit, the factor S from the split Cholesky factorization
   *>          B = S**T*S, as returned by DPBSTF.
   *> \endverbatim
   *>
   *> \param[in] LDBB
   *> \verbatim
   *>          LDBB is INTEGER
   *>          The leading dimension of the array BB.  LDBB >= KB+1.
   *> \endverbatim
   *>
   *> \param[out] W
   *> \verbatim
   *>          W is DOUBLE PRECISION array, dimension (N)
   *>          If INFO = 0, the eigenvalues in ascending order.
   *> \endverbatim
   *>
   *> \param[out] Z
   *> \verbatim
   *>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
   *>          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
   *>          eigenvectors, with the i-th column of Z holding the
   *>          eigenvector associated with W(i).  The eigenvectors are
   *>          normalized so Z**T*B*Z = I.
   *>          If JOBZ = 'N', then Z is not referenced.
   *> \endverbatim
   *>
   *> \param[in] LDZ
   *> \verbatim
   *>          LDZ is INTEGER
   *>          The leading dimension of the array Z.  LDZ >= 1, and if
   *>          JOBZ = 'V', LDZ >= max(1,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.
   *>          If N <= 1,               LWORK >= 1.
   *>          If JOBZ = 'N' and N > 1, LWORK >= 2*N.
   *>          If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*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 LIWORK > 0, IWORK(1) returns the optimal LIWORK.
   *> \endverbatim
   *>
   *> \param[in] LIWORK
   *> \verbatim
   *>          LIWORK is INTEGER
   *>          The dimension of the array IWORK.
   *>          If JOBZ  = 'N' or 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:  if INFO = i, and i is:
   *>             <= N:  the algorithm failed to converge:
   *>                    i off-diagonal elements of an intermediate
   *>                    tridiagonal form did not converge to zero;
   *>             > N:   if INFO = N + i, for 1 <= i <= N, then DPBSTF
   *>                    returned INFO = i: 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 doubleOTHEReigen
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
   *
   *  =====================================================================
       SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W,        SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W,
      $                   Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO )       $                   Z, LDZ, WORK, 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 16 Line 239
      $                   WORK( * ), Z( LDZ, * )       $                   WORK( * ), Z( LDZ, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DSBGVD computes all the eigenvalues, and optionally, the eigenvectors  
 *  of a real generalized symmetric-definite banded eigenproblem, of the  
 *  form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric and  
 *  banded, 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  
 *  =========  
 *  
 *  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.  
 *  
 *  KA      (input) INTEGER  
 *          The number of superdiagonals of the matrix A if UPLO = 'U',  
 *          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.  
 *  
 *  KB      (input) INTEGER  
 *          The number of superdiagonals of the matrix B if UPLO = 'U',  
 *          or the number of subdiagonals if UPLO = 'L'.  KB >= 0.  
 *  
 *  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB, N)  
 *          On entry, the upper or lower triangle of the symmetric band  
 *          matrix A, stored in the first ka+1 rows of the array.  The  
 *          j-th column of A is stored in the j-th column of the array AB  
 *          as follows:  
 *          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;  
 *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).  
 *  
 *          On exit, the contents of AB are destroyed.  
 *  
 *  LDAB    (input) INTEGER  
 *          The leading dimension of the array AB.  LDAB >= KA+1.  
 *  
 *  BB      (input/output) DOUBLE PRECISION array, dimension (LDBB, N)  
 *          On entry, the upper or lower triangle of the symmetric band  
 *          matrix B, stored in the first kb+1 rows of the array.  The  
 *          j-th column of B is stored in the j-th column of the array BB  
 *          as follows:  
 *          if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;  
 *          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).  
 *  
 *          On exit, the factor S from the split Cholesky factorization  
 *          B = S**T*S, as returned by DPBSTF.  
 *  
 *  LDBB    (input) INTEGER  
 *          The leading dimension of the array BB.  LDBB >= KB+1.  
 *  
 *  W       (output) DOUBLE PRECISION array, dimension (N)  
 *          If INFO = 0, the eigenvalues in ascending order.  
 *  
 *  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)  
 *          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of  
 *          eigenvectors, with the i-th column of Z holding the  
 *          eigenvector associated with W(i).  The eigenvectors are  
 *          normalized so Z**T*B*Z = I.  
 *          If JOBZ = 'N', then Z is not referenced.  
 *  
 *  LDZ     (input) INTEGER  
 *          The leading dimension of the array Z.  LDZ >= 1, and if  
 *          JOBZ = 'V', LDZ >= max(1,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.  
 *          If N <= 1,               LWORK >= 1.  
 *          If JOBZ = 'N' and N > 1, LWORK >= 3*N.  
 *          If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*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 LIWORK > 0, IWORK(1) returns the optimal LIWORK.  
 *  
 *  LIWORK  (input) INTEGER  
 *          The dimension of the array IWORK.  
 *          If JOBZ  = 'N' or 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:  if INFO = i, and i is:  
 *             <= N:  the algorithm failed to converge:  
 *                    i off-diagonal elements of an intermediate  
 *                    tridiagonal form did not converge to zero;  
 *             > N:   if INFO = N + i, for 1 <= i <= N, then DPBSTF  
 *                    returned INFO = i: 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  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Line 238 Line 335
       INDWK2 = INDWRK + N*N        INDWK2 = INDWRK + N*N
       LLWRK2 = LWORK - INDWK2 + 1        LLWRK2 = LWORK - INDWK2 + 1
       CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ,        CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ,
      $             WORK( INDWRK ), IINFO )       $             WORK, IINFO )
 *  *
 *     Reduce to tridiagonal form.  *     Reduce to tridiagonal form.
 *  *
Removed from v.1.6  
changed lines
  Added in v.1.19

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