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version 1.5, 2010/08/07 13:18:08 version 1.17, 2016/08/27 15:34:40
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   *> \brief \b DSYGVX
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DSYGVX + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygvx.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygvx.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygvx.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
   *                          VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
   *                          LWORK, IWORK, IFAIL, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBZ, RANGE, UPLO
   *       INTEGER            IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
   *       DOUBLE PRECISION   ABSTOL, VL, VU
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IFAIL( * ), IWORK( * )
   *       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
   *      $                   Z( LDZ, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DSYGVX computes selected eigenvalues, and optionally, 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.
   *> 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 triangle of A and B are stored;
   *>          = 'L':  Lower triangle of A and B are stored.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix pencil (A,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, the lower triangle (if UPLO='L') or the upper
   *>          triangle (if UPLO='U') 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[in] VL
   *> \verbatim
   *>          VL is DOUBLE PRECISION
   *>          If RANGE='V', the lower bound of the interval to
   *>          be searched for eigenvalues. VL < VU.
   *>          Not referenced if RANGE = 'A' or 'I'.
   *> \endverbatim
   *>
   *> \param[in] VU
   *> \verbatim
   *>          VU is DOUBLE PRECISION
   *>          If RANGE='V', the upper bound 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
   *>          If RANGE='I', the index of the
   *>          smallest eigenvalue 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] IU
   *> \verbatim
   *>          IU is INTEGER
   *>          If RANGE='I', the index of the
   *>          largest eigenvalue 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 C to tridiagonal form, where C is the symmetric
   *>          matrix of the standard symmetric problem to which the
   *>          generalized problem is transformed.
   *>
   *>          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 DOUBLE PRECISION array, dimension (LDZ, max(1,M))
   *>          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**T*B*Z = I;
   *>          if ITYPE = 3, Z**T*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 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 length of the array WORK.  LWORK >= max(1,8*N).
   *>          For optimal efficiency, LWORK >= (NB+3)*N,
   *>          where NB is the blocksize for DSYTRD returned by ILAENV.
   *>
   *>          If LWORK = -1, then a workspace query is assumed; the routine
   *>          only calculates the optimal size of the WORK array, returns
   *>          this value as the first entry of the WORK array, and no error
   *>          message related to LWORK is issued by XERBLA.
   *> \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:  DPOTRF or DSYEVX returned an error code:
   *>             <= N:  if INFO = i, DSYEVX 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 June 2016
   *
   *> \ingroup doubleSYeigen
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
   *
   *  =====================================================================
       SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,        SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
      $                   VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,       $                   VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
      $                   LWORK, IWORK, IFAIL, INFO )       $                   LWORK, IWORK, IFAIL, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2.2) --  *  -- LAPACK driver routine (version 3.6.1) --
 *  -- 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..--
 *     June 2010  *     June 2016
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOBZ, RANGE, UPLO        CHARACTER          JOBZ, RANGE, UPLO
Line 18 Line 313
      $                   Z( LDZ, * )       $                   Z( LDZ, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DSYGVX computes selected eigenvalues, and optionally, 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.  
 *  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 triangle of A and B are stored;  
 *          = 'L':  Lower triangle of A and B are stored.  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrix pencil (A,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, the lower triangle (if UPLO='L') or the upper  
 *          triangle (if UPLO='U') 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).  
 *  
 *  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)  
 *          On normal exit, the first M elements contain the selected  
 *          eigenvalues in ascending order.  
 *  
 *  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))  
 *          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**T*B*Z = I;  
 *          if ITYPE = 3, Z**T*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/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))  
 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.  
 *  
 *  LWORK   (input) INTEGER  
 *          The length of the array WORK.  LWORK >= max(1,8*N).  
 *          For optimal efficiency, LWORK >= (NB+3)*N,  
 *          where NB is the blocksize for DSYTRD returned by ILAENV.  
 *  
 *          If LWORK = -1, then a workspace query is assumed; the routine  
 *          only calculates the optimal size of the WORK array, returns  
 *          this value as the first entry of the WORK array, and no error  
 *          message related to LWORK is issued by XERBLA.  
 *  
 *  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:  DPOTRF or DSYEVX returned an error code:  
 *             <= N:  if INFO = i, DSYEVX 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  
 *  
 * =====================================================================  * =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Line 296 Line 427
          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 310 Line 441
          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.1.5  
changed lines
  Added in v.1.17

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