zhpgvx(3)

NAME

ZHPGVX - compute selected eigenvalues and, optionally,

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  ZHPGVX(  ITYPE,  JOBZ, RANGE, UPLO, N, AP, BP,
VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK,  IFAIL,
INFO )
    CHARACTER      JOBZ, RANGE, UPLO
    INTEGER        IL, INFO, ITYPE, IU, LDZ, M, N
    DOUBLE         PRECISION ABSTOL, VL, VU
    INTEGER        IFAIL( * ), IWORK( * )
    DOUBLE         PRECISION RWORK( * ), W( * )
    COMPLEX*16      AP( * ), BP( * ), WORK( * ), Z( LDZ, *
)

PURPOSE

ZHPGVX computes selected eigenvalues and, optionally,

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. Eigen

values and eigenvectors can be selected by specifying either a

range of values or a range of indices for the desired eigenval

ues.

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.

RANGE (input) CHARACTER*1

= 'A': all eigenvalues will be found;
= 'V': all eigenvalues in the half-open interval

(VL,VU] will be found; = 'I': the IL-th through IU-th eigenvalues

will be found.

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 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*16 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.

VL (input) DOUBLE PRECISION

VU (input) DOUBLE PRECISION If RANGE='V', the

lower and upper bounds of the interval to be searched for eigen

values. VL < VU. Not referenced if RANGE = 'A' or 'I'.

IL (input) INTEGER

IU (input) INTEGER If RANGE='I', the indices

(in ascending order) of the smallest and largest eigenvalues to

be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if

N = 0. Not referenced if RANGE = 'A' or 'V'.

ABSTOL (input) DOUBLE PRECISION

The absolute error tolerance for the eigenvalues.

An approximate eigenvalue is accepted as converged when it is de

termined to lie in an interval [a,b] of width less than or equal

to

ABSTOL + EPS * max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is

less than or equal to zero, then EPS*|T| will be used in its

place, where |T| is the 1-norm of the tridiagonal matrix obtained

by reducing AP to tridiagonal form.

Eigenvalues will be computed most accurately when

ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not

zero. If this routine returns with INFO>0, indicating that some

eigenvectors did not converge, try setting ABSTOL to 2*DLAM

CH('S').

M (output) INTEGER

The total number of eigenvalues found. 0 <= M <=

N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W (output) DOUBLE PRECISION array, dimension (N)

On normal exit, the first M elements contain the

selected eigenvalues in ascending order.

Z (output) COMPLEX*16 array, dimension (LDZ, N)

If JOBZ = 'N', then Z is not referenced. If JOBZ

= 'V', then if INFO = 0, the first M columns of Z contain the or

thonormal eigenvectors of the matrix A corresponding to the se

lected eigenvalues, with the i-th column of Z holding the eigen

vector associated with W(i). 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 an eigenvector fails to converge, then that

column of Z contains the latest approximation to the eigenvector,

and the index of the eigenvector is returned in IFAIL. Note: the

user must ensure that at least max(1,M) columns are supplied in

the array Z; if RANGE = 'V', the exact value of M is not known in

advance and an upper bound must be used.

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 (2*N)

RWORK (workspace) DOUBLE PRECISION array, dimension

(7*N)

IWORK (workspace) INTEGER array, dimension (5*N)

IFAIL (output) INTEGER array, dimension (N)

If JOBZ = 'V', then if INFO = 0, the first M ele

ments of IFAIL are zero. If INFO > 0, then IFAIL contains the

indices of the eigenvectors that failed to converge. If JOBZ =

'N', then IFAIL is not referenced.

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value
> 0: ZPPTRF or ZHPEVX returned an error code:
<= N: if INFO = i, ZHPEVX failed to converge; i

eigenvectors failed to converge. Their indices are stored in ar

ray IFAIL. > N: if INFO = N + i, for 1 <= i <= n, then the

leading minor of order i of B is not positive definite. The fac

torization 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

ZHPGVX(3)