sspevx(3F)

NAME
     SSPEVX - compute selected eigenvalues and, optionally, eigenvectors of a
     real symmetric matrix A in packed storage

SYNOPSIS
     SUBROUTINE SSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M,
                        W, Z, LDZ, WORK, IWORK, IFAIL, INFO )

         CHARACTER      JOBZ, RANGE, UPLO

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

         REAL           ABSTOL, VL, VU

         INTEGER        IFAIL( * ), IWORK( * )

         REAL           AP( * ), W( * ), WORK( * ), Z( LDZ, * )

PURPOSE
     SSPEVX computes selected eigenvalues and, optionally, eigenvectors of a
     real symmetric matrix A in packed storage.  Eigenvalues/vectors 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.

     AP      (input/output) REAL array, dimension (N*(N+1)/2)
             On entry, the upper or lower triangle of the symmetric 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, AP is overwritten by values generated during the
             reduction to tridiagonal form.  If UPLO = 'U', the diagonal and
             first superdiagonal of the tridiagonal matrix T overwrite the
             corresponding elements of A, and if UPLO = 'L', the diagonal and
             first subdiagonal of T overwrite the corresponding elements of A.

     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
             AP 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)
             If INFO = 0, the selected eigenvalues in ascending order.

     Z       (output) REAL 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) REAL array, dimension (8*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.