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version 1.8, 2011/07/22 07:38:16 version 1.9, 2011/11/21 20:43:13
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   *> \brief \b ZHPGST
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZHPGVX + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhpgvx.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhpgvx.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhpgvx.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZHPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
   *                          IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
   *                          IWORK, IFAIL, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBZ, RANGE, UPLO
   *       INTEGER            IL, INFO, ITYPE, IU, LDZ, M, N
   *       DOUBLE PRECISION   ABSTOL, VL, VU
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IFAIL( * ), IWORK( * )
   *       DOUBLE PRECISION   RWORK( * ), W( * )
   *       COMPLEX*16         AP( * ), BP( * ), WORK( * ), Z( LDZ, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZHPGVX computes selected eigenvalues and, optionally, eigenvectors
   *> of a complex generalized Hermitian-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 Hermitian, stored in packed format, and B is also
   *> positive definite.  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] 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] 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,out] AP
   *> \verbatim
   *>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
   *>          On entry, the upper or lower triangle of the Hermitian matrix
   *>          A, packed columnwise in a linear array.  The j-th column of A
   *>          is stored in the array AP as follows:
   *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
   *>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
   *>
   *>          On exit, the contents of AP are destroyed.
   *> \endverbatim
   *>
   *> \param[in,out] BP
   *> \verbatim
   *>          BP is COMPLEX*16 array, dimension (N*(N+1)/2)
   *>          On entry, the upper or lower triangle of the Hermitian matrix
   *>          B, packed columnwise in a linear array.  The j-th column of B
   *>          is stored in the array BP as follows:
   *>          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
   *>          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
   *>
   *>          On exit, the triangular factor U or L from the Cholesky
   *>          factorization B = U**H*U or B = L*L**H, in the same storage
   *>          format as B.
   *> \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 AP 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)
   *>          On normal exit, the first M elements contain the selected
   *>          eigenvalues in ascending order.
   *> \endverbatim
   *>
   *> \param[out] Z
   *> \verbatim
   *>          Z is COMPLEX*16 array, dimension (LDZ, N)
   *>          If JOBZ = 'N', then Z is not referenced.
   *>          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).
   *>          The eigenvectors are normalized as follows:
   *>          if ITYPE = 1 or 2, Z**H*B*Z = I;
   *>          if ITYPE = 3, Z**H*inv(B)*Z = 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.
   *>          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 COMPLEX*16 array, dimension (2*N)
   *> \endverbatim
   *>
   *> \param[out] RWORK
   *> \verbatim
   *>          RWORK 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:  ZPPTRF or ZHPEVX returned an error code:
   *>             <= N:  if INFO = i, ZHPEVX failed to converge;
   *>                    i eigenvectors failed to converge.  Their indices
   *>                    are stored in array IFAIL.
   *>             > 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. 
   *
   *> \date November 2011
   *
   *> \ingroup complex16OTHEReigen
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
   *
   *  =====================================================================
       SUBROUTINE ZHPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,        SUBROUTINE ZHPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
      $                   IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,       $                   IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
      $                   IWORK, IFAIL, INFO )       $                   IWORK, IFAIL, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.3.1) --  *  -- 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..--
 *  -- April 2011                                                      --  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOBZ, RANGE, UPLO        CHARACTER          JOBZ, RANGE, UPLO
Line 18 Line 284
       COMPLEX*16         AP( * ), BP( * ), WORK( * ), Z( LDZ, * )        COMPLEX*16         AP( * ), BP( * ), WORK( * ), Z( LDZ, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZHPGVX computes selected eigenvalues and, optionally, eigenvectors  
 *  of a complex generalized Hermitian-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 Hermitian, stored in packed format, and B is also  
 *  positive definite.  Eigenvalues and eigenvectors can be selected by  
 *  specifying either a range of values or a range of indices for the  
 *  desired eigenvalues.  
 *  
 *  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.  
 *  
 *  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.  
 *  
 *  AP      (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)  
 *          On entry, the upper or lower triangle of the Hermitian matrix  
 *          A, packed columnwise in a linear array.  The j-th column of A  
 *          is stored in the array AP as follows:  
 *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;  
 *          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.  
 *  
 *          On exit, the contents of AP are destroyed.  
 *  
 *  BP      (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)  
 *          On entry, the upper or lower triangle of the Hermitian matrix  
 *          B, packed columnwise in a linear array.  The j-th column of B  
 *          is stored in the array BP as follows:  
 *          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;  
 *          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.  
 *  
 *          On exit, the triangular factor U or L from the Cholesky  
 *          factorization B = U**H*U or B = L*L**H, in the same storage  
 *          format as B.  
 *  
 *  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 AP 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)  
 *          On normal exit, the first M elements contain the selected  
 *          eigenvalues in ascending order.  
 *  
 *  Z       (output) COMPLEX*16 array, dimension (LDZ, N)  
 *          If JOBZ = 'N', then Z is not referenced.  
 *          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).  
 *          The eigenvectors are normalized as follows:  
 *          if ITYPE = 1 or 2, Z**H*B*Z = I;  
 *          if ITYPE = 3, Z**H*inv(B)*Z = 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.  
 *          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) COMPLEX*16 array, dimension (2*N)  
 *  
 *  RWORK   (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:  ZPPTRF or ZHPEVX returned an error code:  
 *             <= N:  if INFO = i, ZHPEVX failed to converge;  
 *                    i eigenvectors failed to converge.  Their indices  
 *                    are stored in array IFAIL.  
 *             > 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  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Local Scalars ..  *     .. Local Scalars ..
Removed from v.1.8  
changed lines
  Added in v.1.9

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