NAME

dsbgvd - 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 DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, 
 *      LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
  CHARACTER * 1 JOBZ, UPLO
  INTEGER N, KA, KB, LDAB, LDBB, LDZ, LWORK, LIWORK, INFO
  INTEGER IWORK(*)
  DOUBLE PRECISION AB(LDAB,*), BB(LDBB,*), W(*), Z(LDZ,*), WORK(*)

  SUBROUTINE DSBGVD_64( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, 
 *      Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
  CHARACTER * 1 JOBZ, UPLO
  INTEGER*8 N, KA, KB, LDAB, LDBB, LDZ, LWORK, LIWORK, INFO
  INTEGER*8 IWORK(*)
  DOUBLE PRECISION AB(LDAB,*), BB(LDBB,*), W(*), Z(LDZ,*), WORK(*)

F95 INTERFACE

  SUBROUTINE SBGVD( JOBZ, UPLO, [N], KA, KB, AB, [LDAB], BB, [LDBB], 
 *       W, Z, [LDZ], WORK, [LWORK], [IWORK], [LIWORK], [INFO])
  CHARACTER(LEN=1) :: JOBZ, UPLO
  INTEGER :: N, KA, KB, LDAB, LDBB, LDZ, LWORK, LIWORK, INFO
  INTEGER, DIMENSION(:) :: IWORK
  REAL(8), DIMENSION(:) :: W, WORK
  REAL(8), DIMENSION(:,:) :: AB, BB, Z

  SUBROUTINE SBGVD_64( JOBZ, UPLO, [N], KA, KB, AB, [LDAB], BB, [LDBB], 
 *       W, Z, [LDZ], WORK, [LWORK], [IWORK], [LIWORK], [INFO])
  CHARACTER(LEN=1) :: JOBZ, UPLO
  INTEGER(8) :: N, KA, KB, LDAB, LDBB, LDZ, LWORK, LIWORK, INFO
  INTEGER(8), DIMENSION(:) :: IWORK
  REAL(8), DIMENSION(:) :: W, WORK
  REAL(8), DIMENSION(:,:) :: AB, BB, Z

C INTERFACE

#include <sunperf.h>

void dsbgvd(char jobz, char uplo, int n, int ka, int kb, double *ab, int ldab, double *bb, int ldbb, double *w, double *z, int ldz, double *work, int lwork, int *info);

void dsbgvd_64(char jobz, char uplo, long n, long ka, long kb, double *ab, long ldab, double *bb, long ldbb, double *w, double *z, long ldz, double *work, long lwork, long *info);

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)

 = 'N':  Compute eigenvalues only;

 = 'V':  Compute eigenvalues and eigenvectors.

UPLO (input)

 = 'U':  Upper triangles of A and B are stored;

 = 'L':  Lower triangles of A and B are stored.

N (input)
The order of the matrices A and B. N > = 0.
KA (input)
The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA > = 0.
KB (input)
The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB > = 0.
AB (input/output)
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)
The leading dimension of the array AB. LDAB > = KA+1.
BB (input/output)
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 SPBSTF.
LDBB (input)
The leading dimension of the array BB. LDBB > = KB+1.
W (output)
If INFO = 0, the eigenvalues in ascending order.
Z (output)
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)
The leading dimension of the array Z. LDZ > = 1, and if JOBZ = 'V', LDZ > = max(1,N).
WORK (output)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
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)
On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.
LIWORK (input)
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)

 = 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 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