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NAME
dspevx - compute selected eigenvalues and, optionally,
eigenvectors of a real symmetric matrix A in packed storage
SYNOPSIS
SUBROUTINE DSPEVX( 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
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
#include <sunperf.h>
void dspevx(char jobz, char range, char uplo, int n, double
*dap, double vl, double vu, int il, int iu, double
abstol, int *m, double *w, double *dz, int ldz,
int *ifail, int *info) ;
PURPOSE
DSPEVX computes selected eigenvalues and, optionally, eigen-
vectors 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 eigen-
values.
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) DOUBLE PRECISION array, dimension
(N*(N+1)/2)
On entry, the upper or lower triangle of the sym-
metric 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) 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 obtained by reducing AP to tri-
diagonal 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').
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)
If INFO = 0, the selected eigenvalues 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, then that column of Z con-
tains 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
(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 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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