zpteqr(3)

NAME

ZPTEQR - compute all eigenvalues and, optionally, eigen

vectors of a symmetric positive definite tridiagonal matrix by

first factoring the matrix using DPTTRF and then calling ZBDSQR

to compute the singular values of the bidiagonal factor

SYNOPSIS

SUBROUTINE ZPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
    CHARACTER      COMPZ
    INTEGER        INFO, LDZ, N
    DOUBLE         PRECISION D( * ), E( * ), WORK( * )
    COMPLEX*16     Z( LDZ, * )

PURPOSE

ZPTEQR computes all eigenvalues and, optionally, eigenvec

tors of a symmetric positive definite tridiagonal matrix by first

factoring the matrix using DPTTRF and then calling ZBDSQR to com

pute the singular values of the bidiagonal factor. This routine

computes the eigenvalues of the positive definite tridiagonal ma

trix to high relative accuracy. This means that if the eigenval

ues range over many orders of magnitude in size, then the small

eigenvalues and corresponding eigenvectors will be computed more

accurately than, for example, with the standard QR method.

The eigenvectors of a full or band positive definite Her

mitian matrix can also be found if ZHETRD, ZHPTRD, or ZHBTRD has

been used to reduce this matrix to tridiagonal form. (The reduc

tion to tridiagonal form, however, may preclude the possibility

of obtaining high relative accuracy in the small eigenvalues of

the original matrix, if these eigenvalues range over many orders

of magnitude.)

ARGUMENTS

COMPZ (input) CHARACTER*1

= 'N': Compute eigenvalues only.
= 'V': Compute eigenvectors of original Hermitian

matrix also. Array Z contains the unitary matrix used to reduce

the original matrix to tridiagonal form. = 'I': Compute eigen

vectors of tridiagonal matrix also.

N (input) INTEGER

The order of the matrix. N >= 0.

D (input/output) DOUBLE PRECISION array, dimension

(N)

On entry, the n diagonal elements of the tridiago

nal matrix. On normal exit, D contains the eigenvalues, in de

scending order.

E (input/output) DOUBLE PRECISION array, dimension

(N-1)

On entry, the (n-1) subdiagonal elements of the

tridiagonal matrix. On exit, E has been destroyed.

Z (input/output) COMPLEX*16 array, dimension (LDZ,

N)

On entry, if COMPZ = 'V', the unitary matrix used

in the reduction to tridiagonal form. On exit, if COMPZ = 'V',

the orthonormal eigenvectors of the original Hermitian matrix; if

COMPZ = 'I', the orthonormal eigenvectors of the tridiagonal ma

trix. If INFO > 0 on exit, Z contains the eigenvectors associat

ed with only the stored eigenvalues. If COMPZ = 'N', then Z is

not referenced.

LDZ (input) INTEGER

The leading dimension of the array Z. LDZ >= 1,

and if COMPZ = 'V' or 'I', LDZ >= max(1,N).

WORK (workspace) DOUBLE PRECISION array, dimension

(4*N)

INFO (output) INTEGER

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

gal value.
> 0: if INFO = i, and i is: <= N the Cholesky

factorization of the matrix could not be performed because the i

th principal minor was not positive definite. > N the SVD al

gorithm failed to converge; if INFO = N+i, i off-diagonal ele

ments of the bidiagonal factor did not converge to zero.

LAPACK version 3.0 15 June 2000

ZPTEQR(3)