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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> DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DSBEVX + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbevx.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbevx.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbevx.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, 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, KD, LDAB, LDQ, LDZ, M, N
   *       DOUBLE PRECISION   ABSTOL, VL, VU
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IFAIL( * ), IWORK( * )
   *       DOUBLE PRECISION   AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ),
   *      $                   Z( LDZ, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DSBEVX computes selected eigenvalues and, optionally, eigenvectors
   *> of a real symmetric band matrix A.  Eigenvalues and eigenvectors can
   *> be selected by specifying either 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 triangle of A is stored;
   *>          = 'L':  Lower triangle of A is stored.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix A.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in] KD
   *> \verbatim
   *>          KD is INTEGER
   *>          The number of superdiagonals of the matrix A if UPLO = 'U',
   *>          or the number of subdiagonals if UPLO = 'L'.  KD >= 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 KD+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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
   *>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
   *>
   *>          On exit, AB is overwritten by values generated during the
   *>          reduction to tridiagonal form.  If UPLO = 'U', the first
   *>          superdiagonal and the diagonal of the tridiagonal matrix T
   *>          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
   *>          the diagonal and first subdiagonal of T are returned in the
   *>          first two rows of AB.
   *> \endverbatim
   *>
   *> \param[in] LDAB
   *> \verbatim
   *>          LDAB is INTEGER
   *>          The leading dimension of the array AB.  LDAB >= KD + 1.
   *> \endverbatim
   *>
   *> \param[out] Q
   *> \verbatim
   *>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
   *>          If JOBZ = 'V', the N-by-N orthogonal matrix used in the
   *>                         reduction 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 = 'V', then
   *>          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 AB 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').
   *>
   *>          See "Computing Small Singular Values of Bidiagonal Matrices
   *>          with Guaranteed High Relative Accuracy," by Demmel and
   *>          Kahan, LAPACK Working Note #3.
   *> \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)
   *>          The first M elements contain the selected eigenvalues in
   *>          ascending order.
   *> \endverbatim
   *>
   *> \param[out] Z
   *> \verbatim
   *>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
   *>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
   *>          contain the orthonormal eigenvectors of the matrix A
   *>          corresponding to the selected eigenvalues, with the i-th
   *>          column of Z holding the eigenvector associated with W(i).
   *>          If an eigenvector fails to converge, then that column of Z
   *>          contains the latest approximation to the eigenvector, and the
   *>          index of the eigenvector is returned in IFAIL.
   *>          If JOBZ = 'N', then Z is not referenced.
   *>          Note: the user must ensure that at least max(1,M) columns are
   *>          supplied in the array Z; if RANGE = 'V', the exact value of M
   *>          is not known in advance and an upper bound must be used.
   *> \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 (N)
   *>          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 eigenvectors 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.
   *>          > 0:  if INFO = i, then i eigenvectors failed to converge.
   *>                Their indices are stored in array IFAIL.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleOTHEReigen
   *
   *  =====================================================================
       SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,        SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
      $                   VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,       $                   VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
      $                   IFAIL, INFO )       $                   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 18 Line 274
      $                   Z( LDZ, * )       $                   Z( LDZ, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DSBEVX computes selected eigenvalues and, optionally, eigenvectors  
 *  of a real symmetric band matrix A.  Eigenvalues and eigenvectors can  
 *  be selected by specifying either 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 triangle of A is stored;  
 *          = 'L':  Lower triangle of A is stored.  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrix A.  N >= 0.  
 *  
 *  KD      (input) INTEGER  
 *          The number of superdiagonals of the matrix A if UPLO = 'U',  
 *          or the number of subdiagonals if UPLO = 'L'.  KD >= 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 KD+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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;  
 *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).  
 *  
 *          On exit, AB is overwritten by values generated during the  
 *          reduction to tridiagonal form.  If UPLO = 'U', the first  
 *          superdiagonal and the diagonal of the tridiagonal matrix T  
 *          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',  
 *          the diagonal and first subdiagonal of T are returned in the  
 *          first two rows of AB.  
 *  
 *  LDAB    (input) INTEGER  
 *          The leading dimension of the array AB.  LDAB >= KD + 1.  
 *  
 *  Q       (output) DOUBLE PRECISION array, dimension (LDQ, N)  
 *          If JOBZ = 'V', the N-by-N orthogonal matrix used in the  
 *                         reduction to tridiagonal form.  
 *          If JOBZ = 'N', the array Q is not referenced.  
 *  
 *  LDQ     (input) INTEGER  
 *          The leading dimension of the array Q.  If JOBZ = 'V', then  
 *          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 AB 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').  
 *  
 *          See "Computing Small Singular Values of Bidiagonal Matrices  
 *          with Guaranteed High Relative Accuracy," by Demmel and  
 *          Kahan, LAPACK Working Note #3.  
 *  
 *  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)  
 *          The first M elements contain the selected eigenvalues in  
 *          ascending order.  
 *  
 *  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))  
 *          If JOBZ = 'V', then if INFO = 0, the first M columns of Z  
 *          contain the orthonormal eigenvectors of the matrix A  
 *          corresponding to the selected eigenvalues, with the i-th  
 *          column of Z holding the eigenvector associated with W(i).  
 *          If an eigenvector fails to converge, then that column of Z  
 *          contains the latest approximation to the eigenvector, and the  
 *          index of the eigenvector is returned in IFAIL.  
 *          If JOBZ = 'N', then Z is not referenced.  
 *          Note: the user must ensure that at least max(1,M) columns are  
 *          supplied in the array Z; if RANGE = 'V', the exact value of M  
 *          is not known in advance and an upper bound must be used.  
 *  
 *  LDZ     (input) INTEGER  
 *          The leading dimension of the array Z.  LDZ >= 1, and if  
 *          JOBZ = 'V', LDZ >= max(1,N).  
 *  
 *  WORK    (workspace) DOUBLE PRECISION array, dimension (7*N)  
 *  
 *  IWORK   (workspace) INTEGER array, dimension (5*N)  
 *  
 *  IFAIL   (output) INTEGER array, dimension (N)  
 *          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 eigenvectors 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.  
 *          > 0:  if INFO = i, then i eigenvectors failed to converge.  
 *                Their indices are stored in array IFAIL.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.7  
changed lines
  Added in v.1.8

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