slaed1(3)

NAME

SLAED1 - compute the updated eigensystem of a diagonal ma

trix after modification by a rank-one symmetric matrix

SYNOPSIS

SUBROUTINE SLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK,
IWORK, INFO )
    INTEGER        CUTPNT, INFO, LDQ, N
    REAL           RHO
    INTEGER        INDXQ( * ), IWORK( * )
    REAL           D( * ), Q( LDQ, * ), WORK( * )

PURPOSE

SLAED1 computes the updated eigensystem of a diagonal ma

trix after modification by a rank-one symmetric matrix. This rou

tine is used only for the eigenproblem which requires all eigen

values and eigenvectors of a tridiagonal matrix. SLAED7 handles

the case in which eigenvalues only or eigenvalues and eigenvec

tors of a full symmetric matrix (which was reduced to tridiagonal

form) are desired.

T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) *

D(out) * Q'(out)

where Z = Q'u, u is a vector of length N with ones in

the
CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.

The eigenvectors of the original matrix are stored in

Q, and the
eigenvalues are in D. The algorithm consists of three

stages:

The first stage consists of deflating the size of

the problem
when there are multiple eigenvalues or if there is a

zero in
the Z vector. For each such occurence the dimension

of the
secular equation problem is reduced by one. This

stage is
performed by the routine SLAED2.

The second stage consists of calculating the updated
eigenvalues. This is done by finding the roots of

the secular
equation via the routine SLAED4 (as called by

SLAED3).
This routine also calculates the eigenvectors of the

current
problem.

The final stage consists of computing the updated

eigenvectors
directly using the updated eigenvalues. The eigen

vectors for
the current problem are multiplied with the eigen

vectors from
the overall problem.

ARGUMENTS

N (input) INTEGER

The dimension of the symmetric tridiagonal matrix.

N >= 0.

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

On entry, the eigenvalues of the rank-1-perturbed

matrix. On exit, the eigenvalues of the repaired matrix.

Q (input/output) REAL array, dimension (LDQ,N)

On entry, the eigenvectors of the rank-1-perturbed

matrix. On exit, the eigenvectors of the repaired tridiagonal

matrix.

LDQ (input) INTEGER

The leading dimension of the array Q. LDQ >=

max(1,N).

INDXQ (input/output) INTEGER array, dimension (N)

On entry, the permutation which separately sorts

the two subproblems in D into ascending order. On exit, the per

mutation which will reintegrate the subproblems back into sorted

order, i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.

RHO (input) REAL

The subdiagonal entry used to create the rank-1

modification.

CUTPNT (input) INTEGER The location of the last

eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N/2.

WORK (workspace) REAL array, dimension (4*N + N**2)

IWORK (workspace) INTEGER 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 = 1, an eigenvalue did not converge

FURTHER DETAILS

Based on contributions by

Jeff Rutter, Computer Science Division, University of

California
at Berkeley, USA

Modified by Francoise Tisseur, University of Tennessee.

LAPACK version 3.0 15 June 2000

SLAED1(3)