• NAME
• SYNOPSIS
• F95 INTERFACE
• C INTERFACE
• PURPOSE
• ARGUMENTS
• FURTHER DETAILS

NAME

zhbgvd - compute all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x

SYNOPSIS

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

  SUBROUTINE ZHBGVD_64( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, 
 *      Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
  CHARACTER * 1 JOBZ, UPLO
  DOUBLE COMPLEX AB(LDAB,*), BB(LDBB,*), Z(LDZ,*), WORK(*)
  INTEGER*8 N, KA, KB, LDAB, LDBB, LDZ, LWORK, LRWORK, LIWORK, INFO
  INTEGER*8 IWORK(*)
  DOUBLE PRECISION W(*), RWORK(*)

F95 INTERFACE

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

  SUBROUTINE HBGVD_64( JOBZ, UPLO, [N], KA, KB, AB, [LDAB], BB, [LDBB], 
 *       W, Z, [LDZ], [WORK], [LWORK], [RWORK], [LRWORK], [IWORK], [LIWORK], 
 *       [INFO])
  CHARACTER(LEN=1) :: JOBZ, UPLO
  COMPLEX(8), DIMENSION(:) :: WORK
  COMPLEX(8), DIMENSION(:,:) :: AB, BB, Z
  INTEGER(8) :: N, KA, KB, LDAB, LDBB, LDZ, LWORK, LRWORK, LIWORK, INFO
  INTEGER(8), DIMENSION(:) :: IWORK
  REAL(8), DIMENSION(:) :: W, RWORK

C INTERFACE

#include <sunperf.h>

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

void zhbgvd_64(char jobz, char uplo, long n, long ka, long kb, doublecomplex *ab, long ldab, doublecomplex *bb, long ldbb, double *w, doublecomplex *z, long ldz, long *info);

PURPOSE

zhbgvd computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian 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 Hermitian 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 Hermitian 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(kb+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**H*S, as returned by CPBSTF.

• 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 that Z**H*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 > = N.

• WORK (workspace)
  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 > = N. If JOBZ = 'V' and N > 1, LWORK > = 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.

• RWORK (workspace)
  On exit, if INFO =0, RWORK(1) returns the optimal LRWORK.

• LRWORK (input)
  The dimension of array RWORK. If N < = 1, LRWORK > = 1. If JOBZ = 'N' and N > 1, LRWORK > = N. If JOBZ = 'V' and N > 1, LRWORK > = 1 + 5*N + 2*N**2.

  If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the RWORK array, and no error message related to LRWORK is issued by XERBLA.

• IWORK (workspace)
  On exit, if INFO =0, IWORK(1) returns the optimal LIWORK.

• LIWORK (input)
  The dimension of 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 CPBSTF

  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