clarrv(3)

NAME

CLARRV - compute the eigenvectors of the tridiagonal ma

trix T = L D L^T given L, D and the eigenvalues of L D L^T

SYNOPSIS

SUBROUTINE  CLARRV( N, D, L, ISPLIT, M, W, IBLOCK, GERSCH,
TOL, Z, LDZ, ISUPPZ, WORK, IWORK, INFO )
    INTEGER        INFO, LDZ, M, N
    REAL           TOL
    INTEGER        IBLOCK( * ), ISPLIT( * ), ISUPPZ( *  ),
IWORK( * )
    REAL            D(  *  ), GERSCH( * ), L( * ), W( * ),
WORK( * )
    COMPLEX        Z( LDZ, * )

PURPOSE

CLARRV computes the eigenvectors of the tridiagonal matrix

T = L D L^T given L, D and the eigenvalues of L D L^T. The input

eigenvalues should have high relative accuracy with respect to

the entries of L and D. The desired accuracy of the output can be

specified by the input parameter TOL.

ARGUMENTS

N (input) INTEGER

The order of the matrix. N >= 0.

D (input/output) REAL array, dimension (N)

On entry, the n diagonal elements of the diagonal

matrix D. On exit, D may be overwritten.

L (input/output) REAL array, dimension (N-1)

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

unit bidiagonal matrix L in elements 1 to N-1 of L. L(N) need not

be set. On exit, L is overwritten.

ISPLIT (input) INTEGER array, dimension (N)

The splitting points, at which T breaks up into

submatrices. The first submatrix consists of rows/columns 1 to

ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 through IS

PLIT( 2 ), etc.

TOL (input) REAL

The absolute error tolerance for the eigenval

ues/eigenvectors. Errors in the input eigenvalues must be bound

ed by TOL. The eigenvectors output have residual norms bounded

by TOL, and the dot products between different eigenvectors are

bounded by TOL. TOL must be at least N*EPS*|T|, where EPS is the

machine precision and |T| is the 1-norm of the tridiagonal ma

trix.

M (input) INTEGER

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

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

W (input) REAL array, dimension (N)

The first M elements of W contain the eigenvalues

for which eigenvectors are to be computed. The eigenvalues

should be grouped by split-off block and ordered from smallest to

largest within the block ( The output array W from SLARRE is ex

pected here ). Errors in W must be bounded by TOL (see above).

IBLOCK (input) INTEGER array, dimension (N)

The submatrix indices associated with the corre

sponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs

to the first submatrix from the top, =2 if W(i) belongs to the

second submatrix, etc.

Z (output) COMPLEX array, dimension (LDZ, max(1,M) )

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

columns of Z contain the orthonormal eigenvectors of the matrix T

corresponding to the selected eigenvalues, with the i-th column

of Z holding the eigenvector associated with W(i). If JOBZ =

'N', then Z is not referenced. 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).

ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) )

The support of the eigenvectors in Z, i.e., the

indices indicating the nonzero elements in Z. The i-th eigenvec

tor is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ(

2*i ).

WORK (workspace) REAL array, dimension (13*N)

IWORK (workspace) INTEGER array, dimension (6*N)

INFO (output) INTEGER

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

gal value
> 0: if INFO = 1, internal error in SLARRB if IN

FO = 2, internal error in CSTEIN

FURTHER DETAILS

Based on contributions by

Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of

California at Berkeley, USA

LAPACK version 3.0 15 June 2000

CLARRV(3)