
DSBGVX(l) ) DSBGVX(l)
NAME
DSBGVX  compute selected eigenvalues, and optionally, eigenvectors of a real generalized
symmetricdefinite banded eigenproblem, of the form A*x=(lambda)*B*x
SYNOPSIS
SUBROUTINE DSBGVX( 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
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ), W( * ), WORK( * ),
Z( LDZ, * )
PURPOSE
DSBGVX 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 sub
diagonals if UPLO = 'L'. KA >= 0.
KB (input) INTEGER
The number of superdiagonals of the matrix B if UPLO = 'U', or the number of sub
diagonals if UPLO = 'L'. KB >= 0.
AB (input/output) DOUBLE PRECISION 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 col
umn 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) DOUBLE PRECISION 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 col
umn 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 DPBSTF.
LDBB (input) INTEGER
The leading dimension of the array BB. LDBB >= KB+1.
Q (output) DOUBLE PRECISION 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) DOUBLE PRECISION
VU (input) DOUBLE PRECISION 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) 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 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*DLAMCH('S'), not zero. If this routine returns with INFO>0,
indicating that some eigenvectors did not converge, try setting ABSTOL to
2*DLAMCH('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) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (7N)
IWORK (workspace/output) INTEGER array, dimension (5N)
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.
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 : DPBSTF 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
LAPACK version 3.0 15 June 2000 DSBGVX(l) 
