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NAME
ZHBEVX - compute selected eigenvalues and, optionally,
eigenvectors of a complex Hermitian band matrix A
SYNOPSIS
SUBROUTINE ZHBEVX( 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
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION RWORK( * ), W( * )
COMPLEX*16 AB( LDAB, * ), Q( LDQ, * ), WORK( * ), Z(
LDZ, * )
PURPOSE
ZHBEVX computes selected eigenvalues and, optionally, eigen-
vectors of a complex Hermitian band matrix A.
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.
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*16 array, dimension (LDAB, N)
On entry, the upper or lower triangle of the Hermi-
tian 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 dur-
ing the reduction to tridiagonal form.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KD +
1.
Q (output) COMPLEX*16 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) DOUBLE PRECISION
If RANGE='V', the lower bound of the interval to be
searched for eigenvalues. Not referenced if RANGE =
'A' or 'I'.
VU (input) DOUBLE PRECISION
If RANGE='V', the upper bound of the interval to be
searched for eigenvalues. Not referenced if RANGE =
'A' or 'I'.
IL (input) INTEGER
If RANGE='I', the index (from smallest to largest)
of the smallest eigenvalue to be returned. IL >= 1.
Not referenced if RANGE = 'A' or 'V'.
IU (input) INTEGER
If RANGE='I', the index (from smallest to largest)
of the largest eigenvalue to be returned. min(IL,N)
<= IU <= N. Not referenced 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 AB to tridi-
agonal form.
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) COMPLEX*16 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 corresponding to the selected eigenvalues.
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) COMPLEX*16 array, dimension (N)
RWORK (workspace) DOUBLE PRECISION 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 illegal
value
> 0: if INFO = i, then i eigenvectors failed to
converge. Their indices are stored in array IFAIL.