ssbgvd(3)

NAME

SSBGVD - compute all the eigenvalues, and optionally, the

eigenvectors of a real generalized symmetric-definite 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 generalized symmetric-definite banded

eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are as

sumed to be symmetric and banded, and B is also positive defi

nite. If eigenvectors are desired, it uses a divide and conquer

algorithm.

The divide and conquer algorithm makes very mild assump

tions about floating point arithmetic. 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 X-MP, Cray Y

MP, Cray C-90, or Cray-2. It could conceivably fail on hexadeci

mal 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 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 sym

metric 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.

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 sym

metric 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 fac

torization 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 i-th column of Z holding the

eigenvector associated with W(i). The eigenvectors are normal

ized 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 (LI

WORK)

On exit, if LIWORK > 0, IWORK(1) returns the opti

mal 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 i-th argument had an ille

gal value
> 0: if INFO = i, and i is:
<= N: the algorithm failed to converge: i off-di

agonal elements of an intermediate tridiagonal form did not con

verge to zero; > N: if INFO = N + i, for 1 <= i <= N, then SPB

STF
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 Ken

tucky, USA

LAPACK version 3.0 15 June 2000

SSBGVD(3)