dpteqr(3)

NAME

DPTEQR - compute all eigenvalues and, optionally, eigen

vectors of a symmetric positive definite tridiagonal matrix by

first factoring the matrix using DPTTRF, and then calling DBDSQR

to compute the singular values of the bidiagonal factor

SYNOPSIS

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

PURPOSE

DPTEQR computes all eigenvalues and, optionally, eigenvec

tors of a symmetric positive definite tridiagonal matrix by first

factoring the matrix using DPTTRF, and then calling DBDSQR to

compute the singular values of the bidiagonal factor. This rou

tine computes the eigenvalues of the positive definite tridiago

nal matrix to high relative accuracy. This means that if the

eigenvalues 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 symmetric positive def

inite matrix can also be found if DSYTRD, DSPTRD, or DSBTRD 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 symmetric

matrix also. Array Z contains the orthogonal matrix used to re

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

eigenvectors 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) DOUBLE PRECISION array, dimension

(LDZ, N)

On entry, if COMPZ = 'V', the orthogonal matrix

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

'V', the orthonormal eigenvectors of the original symmetric ma

trix; if COMPZ = 'I', the orthonormal eigenvectors of the tridi

agonal matrix. If INFO > 0 on exit, Z contains the eigenvectors

associated 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

DPTEQR(3)