DSBGVD(3f)

NAME
DSBGVD - 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 DSBGVD(
    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( * )

DOUBLE
    PRECISION AB( LDAB, * ), BB( LDBB, * ), W( * ), WORK( * ), Z( LDZ, * )

PURPOSE
DSBGVD 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 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 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 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 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) 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 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) 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 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 DPBSTF.

LDBB    (input) INTEGER
    The leading dimension of the array BB. LDBB >= KB+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 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.

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 (MAX(1,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 sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.

IWORK   (workspace/output) INTEGER array, dimension (MAX(1,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 sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.

INFO    (output) INTEGER
    = 0: successful exit
    < 0: if INFO = -i, the i-th argument had an illegal value
    > 0: if INFO = i, and i is:
    <= N: the algorithm failed to converge: i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; > N: if INFO = N + i, for 1 <= i <= N, then DPBSTF
    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