NAME

chpgvd - 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 CHPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, 
 *      LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
  CHARACTER * 1 JOBZ, UPLO
  COMPLEX AP(*), BP(*), Z(LDZ,*), WORK(*)
  INTEGER ITYPE, N, LDZ, LWORK, LRWORK, LIWORK, INFO
  INTEGER IWORK(*)
  REAL W(*), RWORK(*)

  SUBROUTINE CHPGVD_64( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, 
 *      LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
  CHARACTER * 1 JOBZ, UPLO
  COMPLEX AP(*), BP(*), Z(LDZ,*), WORK(*)
  INTEGER*8 ITYPE, N, LDZ, LWORK, LRWORK, LIWORK, INFO
  INTEGER*8 IWORK(*)
  REAL W(*), RWORK(*)

F95 INTERFACE

  SUBROUTINE HPGVD( ITYPE, JOBZ, UPLO, [N], AP, BP, W, Z, [LDZ], [WORK], 
 *       [LWORK], [RWORK], [LRWORK], [IWORK], [LIWORK], [INFO])
  CHARACTER(LEN=1) :: JOBZ, UPLO
  COMPLEX, DIMENSION(:) :: AP, BP, WORK
  COMPLEX, DIMENSION(:,:) :: Z
  INTEGER :: ITYPE, N, LDZ, LWORK, LRWORK, LIWORK, INFO
  INTEGER, DIMENSION(:) :: IWORK
  REAL, DIMENSION(:) :: W, RWORK

  SUBROUTINE HPGVD_64( ITYPE, JOBZ, UPLO, [N], AP, BP, W, Z, [LDZ], 
 *       [WORK], [LWORK], [RWORK], [LRWORK], [IWORK], [LIWORK], [INFO])
  CHARACTER(LEN=1) :: JOBZ, UPLO
  COMPLEX, DIMENSION(:) :: AP, BP, WORK
  COMPLEX, DIMENSION(:,:) :: Z
  INTEGER(8) :: ITYPE, N, LDZ, LWORK, LRWORK, LIWORK, INFO
  INTEGER(8), DIMENSION(:) :: IWORK
  REAL, DIMENSION(:) :: W, RWORK

C INTERFACE

#include <sunperf.h>

void chpgvd(int itype, char jobz, char uplo, int n, complex *ap, complex *bp, float *w, complex *z, int ldz, int *info);

void chpgvd_64(long itype, char jobz, char uplo, long n, complex *ap, complex *bp, float *w, complex *z, long ldz, long *info);

PURPOSE

chpgvd 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, stored in packed format, 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.
AP (input/output)
On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1 < =i < =j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j < =i < =n.
On exit, the contents of AP are destroyed.
BP (input/output)
On entry, the upper or lower triangle of the Hermitian matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows: if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1 < =i < =j; if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j < =i < =n.
On exit, the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H, in the same storage format as B.
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. 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 Z is not referenced.
LDZ (input)
The leading dimension of the array Z. LDZ > = 1, and if JOBZ = 'V', LDZ > = max(1,N).
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The dimension of array WORK. If N < = 1, LWORK > = 1. If JOBZ = 'N' and N > 1, LWORK > = N. If JOBZ = 'V' and N > 1, LWORK > = 2*N.
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:  CPPTRF or CHPEVD returned an error code:

 < = N:  if INFO  = i, CHPEVD failed to converge;
i off-diagonal elements of an intermediate
tridiagonal form did not convergeto 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