ZHPGVD(l) ) ZHPGVD(l)
NAME
ZHPGVD - 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 ZHPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )
CHARACTER JOBZ, UPLO
INTEGER INFO, ITYPE, LDZ, LIWORK, LRWORK, LWORK, N
INTEGER IWORK( * )
DOUBLE PRECISION RWORK( * ), W( * )
COMPLEX*16 AP( * ), BP( * ), WORK( * ), Z( LDZ, * )
PURPOSE
ZHPGVD 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) INTEGER
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) 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.
AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
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) COMPLEX*16 array, dimension (N*(N+1)/2)
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) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) COMPLEX*16 array, dimension (LDZ, N)
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) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK (workspace) COMPLEX*16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
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) DOUBLE PRECISION array, dimension (LRWORK)
On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
LRWORK (input) INTEGER
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/output) INTEGER array, dimension (LIWORK)
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK (input) INTEGER
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) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: ZPPTRF or ZHPEVD returned an error code:
<= N: if INFO = i, ZHPEVD 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
LAPACK version 3.0 15 June 2000 ZHPGVD(l)