dlaed7(3)

NAME

DLAED7 - compute the updated eigensystem of a diagonal ma

trix after modification by a rank-one symmetric matrix

SYNOPSIS

SUBROUTINE DLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM,
D, Q, LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, PERM,  GIVPTR, GIVCOL, GIVNUM, WORK, IWORK, INFO )
    INTEGER         CURLVL,  CURPBM, CUTPNT, ICOMPQ, INFO,
LDQ, N, QSIZ, TLVLS
    DOUBLE         PRECISION RHO
    INTEGER        GIVCOL( 2, * ), GIVPTR( * ),  INDXQ(  *
), IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
    DOUBLE          PRECISION  D(  * ), GIVNUM( 2, * ), Q(
LDQ, * ), QSTORE( * ), WORK( * )

PURPOSE

DLAED7 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 optionally eigenvectors of a dense symmetric matrix

that has been reduced to tridiagonal form. DLAED1 handles the

case in which all eigenvalues and eigenvectors of a symmetric

tridiagonal matrix 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 DLAED8.

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

the secular
equation via the routine DLAED4 (as called by

DLAED9).
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

ICOMPQ (input) INTEGER

= 0: Compute eigenvalues only.
= 1: Compute eigenvectors of original dense sym

metric matrix also. On entry, Q contains the orthogonal matrix

used to reduce the original matrix to tridiagonal form.

N (input) INTEGER

The dimension of the symmetric tridiagonal matrix.

N >= 0.

QSIZ (input) INTEGER

The dimension of the orthogonal matrix used to re

duce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ =

1.

TLVLS (input) INTEGER

The total number of merging levels in the overall

divide and conquer tree.

CURLVL (input) INTEGER The current level in the

overall merge routine, 0 <= CURLVL <= TLVLS.

CURPBM (input) INTEGER The current problem in the

current level in the overall merge routine (counting from upper

left to lower right).

D (input/output) DOUBLE PRECISION array, dimension

(N)

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

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

Q (input/output) DOUBLE PRECISION 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 (output) INTEGER array, dimension (N)

The permutation which will reintegrate the subprob

lem just solved back into sorted order, i.e., D( INDXQ( I = 1, N

) ) will be in ascending order.

RHO (input) DOUBLE PRECISION

The subdiagonal element used to create the rank-1

modification.

CUTPNT (input) INTEGER Contains the location of the

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

N.

QSTORE (input/output) DOUBLE PRECISION array, di

mension (N**2+1) Stores eigenvectors of submatrices encountered

during divide and conquer, packed together. QPTR points to begin

ning of the submatrices.

QPTR (input/output) INTEGER array, dimension (N+2)

List of indices pointing to beginning of submatri

ces stored in QSTORE. The submatrices are numbered starting at

the bottom left of the divide and conquer tree, from left to

right and bottom to top.

PRMPTR (input) INTEGER array, dimension (N lg N)

Contains a list of pointers which indicate where in PERM a lev

el's permutation is stored. PRMPTR(i+1) - PRMPTR(i) indicates

the size of the permutation and also the size of the full, non

deflated problem.

PERM (input) INTEGER array, dimension (N lg N)

Contains the permutations (from deflation and sort

ing) to be applied to each eigenblock.

GIVPTR (input) INTEGER array, dimension (N lg N)

Contains a list of pointers which indicate where in GIVCOL a lev

el's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) indi

cates the number of Givens rotations.

GIVCOL (input) INTEGER array, dimension (2, N lg N)

Each pair of numbers indicates a pair of columns to take place in

a Givens rotation.

GIVNUM (input) DOUBLE PRECISION array, dimension

(2, N lg N) Each number indicates the S value to be used in the

corresponding Givens rotation.

WORK (workspace) DOUBLE PRECISION array, dimension

(3*N+QSIZ*N)

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

LAPACK version 3.0 15 June 2000

DLAED7(3)