NAME

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

SYNOPSIS

  SUBROUTINE ZHEGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, 
 *      LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
  CHARACTER * 1 JOBZ, UPLO
  DOUBLE COMPLEX A(LDA,*), B(LDB,*), WORK(*)
  INTEGER ITYPE, N, LDA, LDB, LWORK, LRWORK, LIWORK, INFO
  INTEGER IWORK(*)
  DOUBLE PRECISION W(*), RWORK(*)

  SUBROUTINE ZHEGVD_64( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, 
 *      LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
  CHARACTER * 1 JOBZ, UPLO
  DOUBLE COMPLEX A(LDA,*), B(LDB,*), WORK(*)
  INTEGER*8 ITYPE, N, LDA, LDB, LWORK, LRWORK, LIWORK, INFO
  INTEGER*8 IWORK(*)
  DOUBLE PRECISION W(*), RWORK(*)

F95 INTERFACE

  SUBROUTINE HEGVD( ITYPE, JOBZ, UPLO, [N], A, [LDA], B, [LDB], W, 
 *       [WORK], [LWORK], [RWORK], [LRWORK], [IWORK], [LIWORK], [INFO])
  CHARACTER(LEN=1) :: JOBZ, UPLO
  COMPLEX(8), DIMENSION(:) :: WORK
  COMPLEX(8), DIMENSION(:,:) :: A, B
  INTEGER :: ITYPE, N, LDA, LDB, LWORK, LRWORK, LIWORK, INFO
  INTEGER, DIMENSION(:) :: IWORK
  REAL(8), DIMENSION(:) :: W, RWORK

  SUBROUTINE HEGVD_64( ITYPE, JOBZ, UPLO, [N], A, [LDA], B, [LDB], W, 
 *       [WORK], [LWORK], [RWORK], [LRWORK], [IWORK], [LIWORK], [INFO])
  CHARACTER(LEN=1) :: JOBZ, UPLO
  COMPLEX(8), DIMENSION(:) :: WORK
  COMPLEX(8), DIMENSION(:,:) :: A, B
  INTEGER(8) :: ITYPE, N, LDA, LDB, LWORK, LRWORK, LIWORK, INFO
  INTEGER(8), DIMENSION(:) :: IWORK
  REAL(8), DIMENSION(:) :: W, RWORK

C INTERFACE

#include <sunperf.h>

void zhegvd(int itype, char jobz, char uplo, int n, doublecomplex *a, int lda, doublecomplex *b, int ldb, double *w, int *info);

void zhegvd_64(long itype, char jobz, char uplo, long n, doublecomplex *a, long lda, doublecomplex *b, long ldb, double *w, long *info);

PURPOSE

zhegvd computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian 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

ITYPE (input)
Specifies the problem type to be solved:

 = 1:  A*x  = (lambda)*B*x

 = 2:  A*B*x  = (lambda)*x

 = 3:  B*A*x  = (lambda)*x

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.
A (input/output)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
On exit, if JOBZ = 'V', then if INFO = 0, A contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z = I. If JOBZ = 'N', then on exit the upper triangle (if UPLO ='U') or the lower triangle (if UPLO ='L') of A, including the diagonal, is destroyed.
LDA (input)
The leading dimension of the array A. LDA > = max(1,N).
B (input/output)
On entry, the Hermitian matrix B. If UPLO = 'U', the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = 'L', the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.
On exit, if INFO < = N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H.
LDB (input)
The leading dimension of the array B. LDB > = max(1,N).
W (output)
If INFO = 0, the eigenvalues in ascending order.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The length of the array WORK. If N < = 1, LWORK > = 1. If JOBZ = 'N' and N > 1, LWORK > = N + 1. If JOBZ = 'V' and N > 1, LWORK > = 2*N + 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 the 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 the array IWORK. If N < = 1, LIWORK > = 1. If JOBZ = 'N' and N > 1, LIWORK > = 1. If JOBZ = 'V' and N > 1, LIWORK > = 3 + 5*N.

INFO (output)

 = 0:  successful exit

 < 0:  if INFO  = -i, the i-th argument had an illegal value

 > 0:  CPOTRF or CHEEVD returned an error code:

 < = N:  if INFO  = i, CHEEVD 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 the leading
minor of order i of 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