cheevx(3S)

NAME
     CHEEVX - compute selected eigenvalues and, optionally, eigenvectors of a
     complex Hermitian matrix A

SYNOPSIS
     SUBROUTINE CHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL,
                        M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL, INFO )

         CHARACTER      JOBZ, RANGE, UPLO

         INTEGER        IL, INFO, IU, LDA, LDZ, LWORK, M, N

         REAL           ABSTOL, VL, VU

         INTEGER        IFAIL( * ), IWORK( * )

         REAL           RWORK( * ), W( * )

         COMPLEX        A( LDA, * ), WORK( * ), Z( LDZ, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     CHEEVX computes selected eigenvalues and, optionally, eigenvectors of a
     complex Hermitian 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.

     A       (input/output) COMPLEX array, dimension (LDA, N)
             On entry, the Hermitian 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).

     VL      (input) REAL
             VU      (input) REAL 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) REAL
             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*SLAMCH('S'), not zero.  If
             this routine returns with INFO>0, indicating that some
             eigenvectors did not converge, try setting ABSTOL to
             2*SLAMCH('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) REAL array, dimension (N)
             On normal exit, the first M elements contain the selected
             eigenvalues in ascending order.

     Z       (output) COMPLEX 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/output) COMPLEX array, dimension (LWORK)
             On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

     LWORK   (input) INTEGER
             The length of the array WORK.  LWORK >= max(1,2*N-1).  For
             optimal efficiency, LWORK >= (NB+1)*N, where NB is the max of the
             blocksize for CHETRD and for CUNMTR as 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.

     RWORK   (workspace) REAL 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.
SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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