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NAME
chbevx - compute selected eigenvalues and, optionally,
eigenvectors of a complex Hermitian band matrix A
SYNOPSIS
SUBROUTINE CHBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q,
LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
RWORK, IWORK, IFAIL, INFO )
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
REAL ABSTOL, VL, VU
INTEGER IFAIL( * ), IWORK( * )
REAL RWORK( * ), W( * )
COMPLEX AB( LDAB, * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
#include <sunperf.h>
void chbevx(char jobz, char range, char uplo, int n, int kd,
complex *cab, int ldab, complex *q, int ldq, float
vl, float vu, int il, int iu, float abstol, int
*m, float *w, complex *cz, int ldz, int *ifail,
int *info) ;
PURPOSE
CHBEVX computes selected eigenvalues and, optionally, eigen-
vectors of a complex Hermitian band matrix A. Eigenvalues
and eigenvectors 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.
KD (input) INTEGER
The number of superdiagonals of the matrix A if
UPLO = 'U', or the number of subdiagonals if UPLO
= 'L'. KD >= 0.
AB (input/output) COMPLEX array, dimension (LDAB, N)
On entry, the upper or lower triangle of the Her-
mitian band matrix A, stored in the first KD+1
rows of the array. The j-th column of A is stored
in the j-th column of the array AB as follows: if
UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-
kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j)
for j<=i<=min(n,j+kd).
On exit, AB is overwritten by values generated
during the reduction to tridiagonal form.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KD
+ 1.
Q (output) COMPLEX array, dimension (LDQ, N)
If JOBZ = 'V', the N-by-N unitary matrix used in
the reduction to tridiagonal form. If JOBZ = 'N',
the array Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. If JOBZ =
'V', then LDQ >= 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 refer-
enced 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 AB to tri-
diagonal 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)
The first M elements contain the selected eigen-
values 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 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) COMPLEX array, dimension (N)
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 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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