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 symmetricdefinite 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

= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.  RANGE (input) CHARACTER*1

= 'A': all eigenvalues will be found.
= 'V': all eigenvalues in the halfopen interval (VL,VU] will be found. = 'I': the ILth through IUth eigenvalues will be found.  UPLO (input) CHARACTER*1

= 'U': Upper triangles of A and B are stored;
= 'L': 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 = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.
 KB (input) INTEGER
 The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. 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 jth column of A is stored in the jth column of the array AB as follows: if UPLO = 'U', AB(ka+1+ij,j) = A(i,j) for max(1,jka)<=i<=j; if UPLO = 'L', AB(1+ij,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 jth column of B is stored in the jth column of the array BB as follows: if UPLO = 'U', BB(ka+1+ij,j) = B(i,j) for max(1,jkb)<=i<=j; if UPLO = 'L', BB(1+ij,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 = 'V', the nbyn 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.
 LDQ (input) INTEGER
 The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1. If JOBZ = 'V', 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 referenced 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 1norm 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').
 M (output) INTEGER
 The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IUIL+1.
 W (output) REAL array, dimension (N)
 If INFO = 0, the eigenvalues in ascending order.
 Z (output) REAL array, dimension (LDZ, N)
 If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the ith 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.
 LDZ (input) INTEGER
 The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', 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 = '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.
 INFO (output) INTEGER

= 0 : successful exit
< 0 : if INFO = i, the ith 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 byMark Fahey, Department of Mathematics, Univ. of Kentucky, USA