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.TH SSPEVX l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
SSPEVX - compute selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage
.SH SYNOPSIS
.TP 19
SUBROUTINE SSPEVX(
JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
INFO )
.TP 19
.ti +4
CHARACTER
JOBZ, RANGE, UPLO
.TP 19
.ti +4
INTEGER
IL, INFO, IU, LDZ, M, N
.TP 19
.ti +4
REAL
ABSTOL, VL, VU
.TP 19
.ti +4
INTEGER
IFAIL( * ), IWORK( * )
.TP 19
.ti +4
REAL
AP( * ), W( * ), WORK( * ), Z( LDZ, * )
.SH 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.
.br
.SH ARGUMENTS
.TP 8
JOBZ (input) CHARACTER*1
= 'N': Compute eigenvalues only;
.br
= 'V': Compute eigenvalues and eigenvectors.
.TP 8
RANGE (input) CHARACTER*1
.br
= 'A': all eigenvalues will be found;
.br
= 'V': all eigenvalues in the half-open interval (VL,VU]
will be found;
= 'I': the IL-th through IU-th eigenvalues will be found.
.TP 8
UPLO (input) CHARACTER*1
.br
= 'U': Upper triangle of A is stored;
.br
= 'L': Lower triangle of A is stored.
.TP 8
N (input) INTEGER
The order of the matrix A. N >= 0.
.TP 8
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.
.TP 8
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'.
.TP 8
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'.
.TP 8
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.
.TP 8
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.
.TP 8
W (output) REAL array, dimension (N)
If INFO = 0, the selected eigenvalues in ascending order.
.TP 8
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.
.TP 8
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = 'V', LDZ >= max(1,N).
.TP 8
WORK (workspace) REAL array, dimension (8*N)
.TP 8
IWORK (workspace) INTEGER array, dimension (5*N)
.TP 8
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.
.TP 8
INFO (output) INTEGER
= 0: successful exit
.br
< 0: if INFO = -i, the i-th argument had an illegal value
.br
> 0: if INFO = i, then i eigenvectors failed to converge.
Their indices are stored in array IFAIL.
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