chpgvd(3)

NAME

CHPGVD - compute all the eigenvalues and, optionally, the

eigenvectors of a complex generalized Hermitian-definite eigen

problem, 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      JOBZ, UPLO
    INTEGER         INFO,  ITYPE,  LDZ,  LIWORK,   LRWORK,
LWORK, N
    INTEGER        IWORK( * )
    REAL           RWORK( * ), W( * )
    COMPLEX         AP( * ), BP( * ), WORK( * ), Z( LDZ, *
)

PURPOSE

CHPGVD computes all the eigenvalues and, optionally, the

eigenvectors of a complex generalized Hermitian-definite eigen

problem, 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 assump

tions 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 hexadeci

mal 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 array, dimension

(N*(N+1)/2)

On entry, the upper or lower triangle of the Her

mitian 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 array, dimension

(N*(N+1)/2)

On entry, the upper or lower triangle of the Her

mitian 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) REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

Z (output) COMPLEX 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 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) REAL 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 (LI

WORK)

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 ille

gal 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 Ken

tucky, USA

LAPACK version 3.0 15 June 2000

CHPGVD(3)