
SSBGVD(l) ) SSBGVD(l)
NAME
SSBGVD  compute all the eigenvalues, and optionally, the eigenvectors of a real general
ized symmetricdefinite banded eigenproblem, of the form A*x=(lambda)*B*x
SYNOPSIS
SUBROUTINE SSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, LWORK,
IWORK, LIWORK, INFO )
CHARACTER JOBZ, UPLO
INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N
INTEGER IWORK( * )
REAL AB( LDAB, * ), BB( LDBB, * ), W( * ), WORK( * ), Z( LDZ, * )
PURPOSE
SSBGVD computes all the eigenvalues, and optionally, the eigenvectors of a real general
ized 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. If eigenvectors
are desired, it uses a divide and conquer algorithm.
The divide and conquer algorithm makes very mild assumptions about floating point arith
metic. It will work on machines with a guard digit in add/subtract, or on those binary
machines without guard digits which subtract like the Cray XMP, Cray YMP, Cray C90, or
Cray2. It could conceivably fail on hexadecimal or decimal machines without guard digits,
but we know of none.
ARGUMENTS
JOBZ (input) CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
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) 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 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) 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 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 SPBSTF.
LDBB (input) INTEGER
The leading dimension of the array BB. LDBB >= KB+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 (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. If N <= 1, LWORK >= 1. If JOBZ =
'N' and N > 1, LWORK >= 3*N. If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2.
If LWORK = 1, then a workspace query is assumed; the routine only calculates the
optimal size of the WORK array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued by XERBLA.
IWORK (workspace/output) INTEGER array, dimension (LIWORK)
On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.
LIWORK (input) INTEGER
The dimension of the array IWORK. If JOBZ = 'N' or N <= 1, LIWORK >= 1. If JOBZ
= 'V' and N > 1, LIWORK >= 3 + 5*N.
If LIWORK = 1, then a workspace query is assumed; the routine only calculates the
optimal size of the IWORK array, returns this value as the first entry of the
IWORK array, and no error message related to LIWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
> 0: if INFO = i, and i is:
<= N: the algorithm failed to converge: i offdiagonal elements of an intermedi
ate tridiagonal form did not converge to zero; > N: if INFO = N + i, for 1 <= i
<= N, then SPBSTF
returned INFO = i: 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 SSBGVD(l) 
