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NAME
dstevx - compute selected eigenvalues and, optionally,
eigenvectors of a real symmetric tridiagonal matrix A
SYNOPSIS
SUBROUTINE DSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
CHARACTER JOBZ, RANGE
INTEGER IL, INFO, IU, LDZ, M, N
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ,
* )
#include <sunperf.h>
void dstevx(char jobz, char range, int n, double *d, double
*e, double vl, double vu, int il, int iu, double
abstol, int *m, double *w, double *dz, int ldz,
int *ifail, int *info) ;
PURPOSE
DSTEVX computes selected eigenvalues and, optionally, eigen-
vectors of a real symmetric tridiagonal matrix A. Eigen-
values 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.
N (input) INTEGER
The order of the matrix. N >= 0.
D (input/output) DOUBLE PRECISION array, dimension
(N)
On entry, the n diagonal elements of the tridiago-
nal matrix A. On exit, D may be multiplied by a
constant factor chosen to avoid over/underflow in
computing the eigenvalues.
E (input/output) DOUBLE PRECISION array, dimension
(N)
On entry, the (n-1) subdiagonal elements of the
tridiagonal matrix A in elements 1 to N-1 of E;
E(N) need not be set. On exit, E may be multi-
plied by a constant factor chosen to avoid
over/underflow in computing the eigenvalues.
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 refer-
enced 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.
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').
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) DOUBLE PRECISION array, dimension (N)
The first M elements contain the selected eigen-
values in ascending order.
Z (output) DOUBLE PRECISION 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 eigenvec-
tor fails to converge (INFO > 0), 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) DOUBLE PRECISION array, dimension
(5*N)
IWORK (workspace) INTEGER array, dimension (5*N)
IFAIL (output) INTEGER array, dimension (N)
If JOBZ = 'V', then if INFO = 0, the first M ele-
ments 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 ille-
gal value
> 0: if INFO = i, then i eigenvectors failed to
converge. Their indices are stored in array
IFAIL.
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