.TH SSBGVX l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
SSBGVX - compute selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x
.SH SYNOPSIS
.TP 19
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 )
.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
AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
W( * ), WORK( * ), Z( LDZ, * )
.SH 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.

.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) 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 = '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) 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 = 'U', BB(ka+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**T*S, as returned by SPBSTF.
.TP 8
LDBB    (input) INTEGER
The leading dimension of the array BB.  LDBB >= KB+1.
.TP 8
Q       (output) REAL 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 A 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) REAL 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 Z**T*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 >= max(1,N).
.TP 8
WORK    (workspace/output) REAL array, dimension (7N)
.TP 8
IWORK   (workspace/output) INTEGER array, dimension (5N)
.TP 8
IFAIL   (input) INTEGER array, dimension (M)
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 eigenvalues 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
<= 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.
.SH FURTHER DETAILS
Based on contributions by
.br
   Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA