# ssbgvx (l) - Linux Manuals

## NAME

SSBGVX - computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x

## SYNOPSIS

SUBROUTINE SSBGVX(
JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )

CHARACTER JOBZ, RANGE, UPLO

INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M, N

REAL ABSTOL, VL, VU

INTEGER IFAIL( * ), IWORK( * )

REAL AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ), W( * ), WORK( * ), Z( LDZ, * )

## PURPOSE

SSBGVX computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric 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.

## ARGUMENTS

JOBZ (input) CHARACTER*1
= aqNaq: Compute eigenvalues only;
= aqVaq: Compute eigenvalues and eigenvectors.
RANGE (input) CHARACTER*1

= aqAaq: all eigenvalues will be found.
= aqVaq: all eigenvalues in the half-open interval (VL,VU] will be found. = aqIaq: the IL-th through IU-th eigenvalues will be found.
UPLO (input) CHARACTER*1

= aqUaq: Upper triangles of A and B are stored;
= aqLaq: Lower triangles of A and B are stored.
N (input) INTEGER
The order of the matrices A and B. N >= 0.
KA (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = aqUaq, or the number of subdiagonals if UPLO = aqLaq. KA >= 0.
KB (input) INTEGER
The number of superdiagonals of the matrix B if UPLO = aqUaq, or the number of subdiagonals if UPLO = aqLaq. KB >= 0.
AB (input/output) REAL array, dimension (LDAB, N)
On entry, the upper or lower triangle of the symmetric 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 = aqUaq, AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = aqLaq, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KA+1.
BB (input/output) REAL array, dimension (LDBB, N)
On entry, the upper or lower triangle of the symmetric 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 = aqUaq, BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; if UPLO = aqLaq, 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**T*S, as returned by SPBSTF.
LDBB (input) INTEGER
The leading dimension of the array BB. LDBB >= KB+1.
Q (output) REAL array, dimension (LDQ, N)
If JOBZ = aqVaq, 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 = aqNaq, the array Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. If JOBZ = aqNaq, LDQ >= 1. If JOBZ = aqVaq, LDQ >= max(1,N).
VL (input) REAL
VU (input) REAL If RANGE=aqVaq, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = aqAaq or aqIaq.
IL (input) INTEGER
IU (input) INTEGER If RANGE=aqIaq, 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 = aqAaq or aqVaq.
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 A to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH(aqSaq), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH(aqSaq).
M (output) INTEGER
The total number of eigenvalues found. 0 <= M <= N. If RANGE = aqAaq, M = N, and if RANGE = aqIaq, M = IU-IL+1.
W (output) REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) REAL array, dimension (LDZ, N)
If JOBZ = aqVaq, 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 Z**T*B*Z = I. If JOBZ = aqNaq, then Z is not referenced.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if JOBZ = aqVaq, LDZ >= max(1,N).
WORK (workspace/output) REAL array, dimension (7N)
IWORK (workspace/output) INTEGER array, dimension (5N)
IFAIL (output) INTEGER array, dimension (M)
If JOBZ = aqVaq, then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvalues that failed to converge. If JOBZ = aqNaq, then IFAIL is not referenced.
INFO (output) INTEGER
= 0 : successful exit
< 0 : if INFO = -i, the i-th argument had an illegal value
<= N: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in IFAIL. > N : SPBSTF returned an error code; i.e., if INFO = N + i, for 1 <= i <= N, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.

## FURTHER DETAILS

Based on contributions by

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA