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.TH CHBGVX l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
CHBGVX - compute all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x
.SH SYNOPSIS
.TP 19
SUBROUTINE CHBGVX(
JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
LDZ, WORK, RWORK, IWORK, IFAIL, INFO )
.TP 19
.ti +4
CHARACTER
JOBZ, RANGE, UPLO
.TP 19
.ti +4
INTEGER
IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
N
.TP 19
.ti +4
REAL
ABSTOL, VL, VU
.TP 19
.ti +4
INTEGER
IFAIL( * ), IWORK( * )
.TP 19
.ti +4
REAL
RWORK( * ), W( * )
.TP 19
.ti +4
COMPLEX
AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
WORK( * ), Z( LDZ, * )
.SH PURPOSE
CHBGVX computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian and banded, and B is also positive definite. Eigenvalues and
eigenvectors can be selected by specifying either all eigenvalues,
a range of values or a range of indices for the desired eigenvalues.
.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 triangles of A and B are stored;
.br
= 'L': Lower triangles of A and B are stored.
.TP 8
N (input) INTEGER
The order of the matrices A and B. N >= 0.
.TP 8
KA (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KA >= 0.
.TP 8
KB (input) INTEGER
The number of superdiagonals of the matrix B if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KB >= 0.
.TP 8
AB (input/output) COMPLEX array, dimension (LDAB, N)
On entry, the upper or lower triangle of the Hermitian band
matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).
On exit, the contents of AB are destroyed.
.TP 8
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KA+1.
.TP 8
BB (input/output) COMPLEX array, dimension (LDBB, N)
On entry, the upper or lower triangle of the Hermitian band
matrix B, stored in the first kb+1 rows of the array. The
j-th column of B is stored in the j-th column of the array BB
as follows:
if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb).
On exit, the factor S from the split Cholesky factorization
B = S**H*S, as returned by CPBSTF.
.TP 8
LDBB (input) INTEGER
The leading dimension of the array BB. LDBB >= KB+1.
.TP 8
Q (output) COMPLEX array, dimension (LDQ, N)
If JOBZ = 'V', the n-by-n matrix used in the reduction of
A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
and consequently C to tridiagonal form.
If JOBZ = 'N', the array Q is not referenced.
.TP 8
LDQ (input) INTEGER
The leading dimension of the array Q. If JOBZ = 'N',
LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).
.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').
.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 eigenvalues in ascending order.
.TP 8
Z (output) COMPLEX array, dimension (LDZ, N)
If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
eigenvectors, with the i-th column of Z holding the
eigenvector associated with W(i). The eigenvectors are
normalized so that Z**H*B*Z = I.
If JOBZ = 'N', then Z is not referenced.
.TP 8
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = 'V', LDZ >= N.
.TP 8
WORK (workspace) COMPLEX array, dimension (N)
.TP 8
RWORK (workspace) REAL array, dimension (7*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, and i is:
.br
<= N: then i eigenvectors failed to converge. Their
indices are stored in array IFAIL.
> N: if INFO = N + i, for 1 <= i <= N, then CPBSTF
.br
returned INFO = i: B is not positive definite.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.
.SH FURTHER DETAILS
Based on contributions by
.br
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
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