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version 1.7, 2010/12/21 13:53:37 version 1.8, 2011/11/21 20:43:03
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   *> \brief \b DSBGST
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DSBGVX + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbgvx.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbgvx.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbgvx.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
   *                          LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
   *                          LDZ, WORK, IWORK, IFAIL, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBZ, RANGE, UPLO
   *       INTEGER            IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
   *      $                   N
   *       DOUBLE PRECISION   ABSTOL, VL, VU
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IFAIL( * ), IWORK( * )
   *       DOUBLE PRECISION   AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
   *      $                   W( * ), WORK( * ), Z( LDZ, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DSBGVX computes selected eigenvalues, and optionally, 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.  Eigenvalues and
   *> eigenvectors can be selected by specifying either all eigenvalues,
   *> a range of values or a range of indices for the desired eigenvalues.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOBZ
   *> \verbatim
   *>          JOBZ is CHARACTER*1
   *>          = 'N':  Compute eigenvalues only;
   *>          = 'V':  Compute eigenvalues and eigenvectors.
   *> \endverbatim
   *>
   *> \param[in] RANGE
   *> \verbatim
   *>          RANGE is CHARACTER*1
   *>          = 'A': all eigenvalues will be found.
   *>          = 'V': all eigenvalues in the half-open interval (VL,VU]
   *>                 will be found.
   *>          = 'I': the IL-th through IU-th eigenvalues will be found.
   *> \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] Q
   *> \verbatim
   *>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
   *>          If JOBZ = 'V', the n-by-n matrix used in the reduction of
   *>          A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
   *>          and consequently C to tridiagonal form.
   *>          If JOBZ = 'N', the array Q is not referenced.
   *> \endverbatim
   *>
   *> \param[in] LDQ
   *> \verbatim
   *>          LDQ is INTEGER
   *>          The leading dimension of the array Q.  If JOBZ = 'N',
   *>          LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in] VL
   *> \verbatim
   *>          VL is DOUBLE PRECISION
   *> \endverbatim
   *>
   *> \param[in] VU
   *> \verbatim
   *>          VU is DOUBLE PRECISION
   *>
   *>          If RANGE='V', the lower and upper bounds of the interval to
   *>          be searched for eigenvalues. VL < VU.
   *>          Not referenced if RANGE = 'A' or 'I'.
   *> \endverbatim
   *>
   *> \param[in] IL
   *> \verbatim
   *>          IL is INTEGER
   *> \endverbatim
   *>
   *> \param[in] IU
   *> \verbatim
   *>          IU is INTEGER
   *>
   *>          If RANGE='I', the indices (in ascending order) of the
   *>          smallest and largest eigenvalues to be returned.
   *>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
   *>          Not referenced if RANGE = 'A' or 'V'.
   *> \endverbatim
   *>
   *> \param[in] ABSTOL
   *> \verbatim
   *>          ABSTOL is DOUBLE PRECISION
   *>          The absolute error tolerance for the eigenvalues.
   *>          An approximate eigenvalue is accepted as converged
   *>          when it is determined to lie in an interval [a,b]
   *>          of width less than or equal to
   *>
   *>                  ABSTOL + EPS *   max( |a|,|b| ) ,
   *>
   *>          where EPS is the machine precision.  If ABSTOL is less than
   *>          or equal to zero, then  EPS*|T|  will be used in its place,
   *>          where |T| is the 1-norm of the tridiagonal matrix obtained
   *>          by reducing A to tridiagonal form.
   *>
   *>          Eigenvalues will be computed most accurately when ABSTOL is
   *>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
   *>          If this routine returns with INFO>0, indicating that some
   *>          eigenvectors did not converge, try setting ABSTOL to
   *>          2*DLAMCH('S').
   *> \endverbatim
   *>
   *> \param[out] M
   *> \verbatim
   *>          M is INTEGER
   *>          The total number of eigenvalues found.  0 <= M <= N.
   *>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+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 (7*N)
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (5*N)
   *> \endverbatim
   *>
   *> \param[out] IFAIL
   *> \verbatim
   *>          IFAIL is INTEGER array, dimension (M)
   *>          If JOBZ = 'V', then if INFO = 0, the first M elements of
   *>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
   *>          indices of the eigenvalues that failed to converge.
   *>          If JOBZ = 'N', then IFAIL is 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
   *>          <= N: if INFO = i, then i eigenvectors failed to converge.
   *>                  Their indices are stored in IFAIL.
   *>          > N : DPBSTF returned an error code; i.e.,
   *>                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. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleOTHEReigen
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
   *
   *  =====================================================================
       SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,        SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
      $                   LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,       $                   LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
      $                   LDZ, WORK, IWORK, IFAIL, INFO )       $                   LDZ, WORK, IWORK, IFAIL, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2) --  *  -- 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..--
 *     November 2006  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOBZ, RANGE, UPLO        CHARACTER          JOBZ, RANGE, UPLO
Line 19 Line 302
      $                   W( * ), WORK( * ), Z( LDZ, * )       $                   W( * ), WORK( * ), Z( LDZ, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DSBGVX computes selected eigenvalues, and optionally, 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.  Eigenvalues and  
 *  eigenvectors can be selected by specifying either all eigenvalues,  
 *  a range of values or a range of indices for the desired eigenvalues.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  JOBZ    (input) CHARACTER*1  
 *          = 'N':  Compute eigenvalues only;  
 *          = 'V':  Compute eigenvalues and eigenvectors.  
 *  
 *  RANGE   (input) CHARACTER*1  
 *          = 'A': all eigenvalues will be found.  
 *          = 'V': all eigenvalues in the half-open interval (VL,VU]  
 *                 will be found.  
 *          = 'I': the IL-th through IU-th eigenvalues will be found.  
 *  
 *  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.  
 *  
 *  Q       (output) DOUBLE PRECISION array, dimension (LDQ, N)  
 *          If JOBZ = 'V', the n-by-n matrix used in the reduction of  
 *          A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,  
 *          and consequently C to tridiagonal form.  
 *          If JOBZ = 'N', the array Q is not referenced.  
 *  
 *  LDQ     (input) INTEGER  
 *          The leading dimension of the array Q.  If JOBZ = 'N',  
 *          LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).  
 *  
 *  VL      (input) DOUBLE PRECISION  
 *  VU      (input) DOUBLE PRECISION  
 *          If RANGE='V', the lower and upper bounds of the interval to  
 *          be searched for eigenvalues. VL < VU.  
 *          Not referenced if RANGE = 'A' or 'I'.  
 *  
 *  IL      (input) INTEGER  
 *  IU      (input) INTEGER  
 *          If RANGE='I', the indices (in ascending order) of the  
 *          smallest and largest eigenvalues to be returned.  
 *          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.  
 *          Not referenced if RANGE = 'A' or 'V'.  
 *  
 *  ABSTOL  (input) DOUBLE PRECISION  
 *          The absolute error tolerance for the eigenvalues.  
 *          An approximate eigenvalue is accepted as converged  
 *          when it is determined to lie in an interval [a,b]  
 *          of width less than or equal to  
 *  
 *                  ABSTOL + EPS *   max( |a|,|b| ) ,  
 *  
 *          where EPS is the machine precision.  If ABSTOL is less than  
 *          or equal to zero, then  EPS*|T|  will be used in its place,  
 *          where |T| is the 1-norm of the tridiagonal matrix obtained  
 *          by reducing A to tridiagonal form.  
 *  
 *          Eigenvalues will be computed most accurately when ABSTOL is  
 *          set to twice the underflow threshold 2*DLAMCH('S'), not zero.  
 *          If this routine returns with INFO>0, indicating that some  
 *          eigenvectors did not converge, try setting ABSTOL to  
 *          2*DLAMCH('S').  
 *  
 *  M       (output) INTEGER  
 *          The total number of eigenvalues found.  0 <= M <= N.  
 *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+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 (7*N)  
 *  
 *  IWORK   (workspace/output) INTEGER array, dimension (5*N)  
 *  
 *  IFAIL   (output) INTEGER array, dimension (M)  
 *          If JOBZ = 'V', then if INFO = 0, the first M elements of  
 *          IFAIL are zero.  If INFO > 0, then IFAIL contains the  
 *          indices of the eigenvalues that failed to converge.  
 *          If JOBZ = 'N', then IFAIL is not referenced.  
 *  
 *  INFO    (output) INTEGER  
 *          = 0 : successful exit  
 *          < 0 : if INFO = -i, the i-th argument had an illegal value  
 *          <= N: if INFO = i, then i eigenvectors failed to converge.  
 *                  Their indices are stored in IFAIL.  
 *          > N : DPBSTF returned an error code; i.e.,  
 *                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  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.7  
changed lines
  Added in v.1.8

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