slaed8(3)

NAME

SLAED8 - merge the two sets of eigenvalues together into a

single sorted set

SYNOPSIS

SUBROUTINE  SLAED8(  ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ,
RHO, CUTPNT, Z,  DLAMDA,  Q2,  LDQ2,  W,  PERM,  GIVPTR,  GIVCOL,
GIVNUM, INDXP, INDX, INFO )
    INTEGER         CUTPNT,  GIVPTR, ICOMPQ, INFO, K, LDQ,
LDQ2, N, QSIZ
    REAL           RHO
    INTEGER        GIVCOL( 2, * ), INDX( * ), INDXP( *  ),
INDXQ( * ), PERM( * )
    REAL           D( * ), DLAMDA( * ), GIVNUM( 2, * ), Q(
LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )

PURPOSE

SLAED8 merges the two sets of eigenvalues together into a

single sorted set. Then it tries to deflate the size of the prob

lem. There are two ways in which deflation can occur: when two

or more eigenvalues are close together or if there is a tiny ele

ment in the Z vector. For each such occurrence the order of the

related secular equation problem is reduced by one.

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.

K (output) INTEGER

The number of non-deflated eigenvalues, and the or

der of the related secular equation.

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.

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

On entry, the eigenvalues of the two submatrices to

be combined. On exit, the trailing (N-K) updated eigenvalues

(those which were deflated) sorted into increasing order.

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

If ICOMPQ = 0, Q is not referenced. Otherwise, on

entry, Q contains the eigenvectors of the partially solved system

which has been previously updated in matrix multiplies with other

partially solved eigensystems. On exit, Q contains the trailing

(N-K) updated eigenvectors (those which were deflated) in its

last N-K columns.

LDQ (input) INTEGER

The leading dimension of the array Q. LDQ >=

max(1,N).

INDXQ (input) INTEGER array, dimension (N)

The permutation which separately sorts the two sub

problems in D into ascending order. Note that elements in the

second half of this permutation must first have CUTPNT added to

their values in order to be accurate.

RHO (input/output) REAL

On entry, the off-diagonal element associated with

the rank-1 cut which originally split the two submatrices which

are now being recombined. On exit, RHO has been modified to the

value required by SLAED3.

CUTPNT (input) INTEGER The location of the last

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

Z (input) REAL array, dimension (N)

On entry, Z contains the updating vector (the last

row of the first sub-eigenvector matrix and the first row of the

second sub-eigenvector matrix). On exit, the contents of Z are

destroyed by the updating process.

DLAMDA (output) REAL array, dimension (N) A copy of

the first K eigenvalues which will be used by SLAED3 to form the

secular equation.

Q2 (output) REAL array, dimension (LDQ2,N)

If ICOMPQ = 0, Q2 is not referenced. Otherwise, a

copy of the first K eigenvectors which will be used by SLAED7 in

a matrix multiply (SGEMM) to update the new eigenvectors.

LDQ2 (input) INTEGER

The leading dimension of the array Q2. LDQ2 >=

max(1,N).

W (output) REAL array, dimension (N)

The first k values of the final deflation-altered

z-vector and will be passed to SLAED3.

PERM (output) INTEGER array, dimension (N)

The permutations (from deflation and sorting) to be

applied to each eigenblock.

GIVPTR (output) INTEGER The number of Givens rota

tions which took place in this subproblem.

GIVCOL (output) INTEGER array, dimension (2, N)

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

a Givens rotation.

GIVNUM (output) REAL array, dimension (2, N) Each

number indicates the S value to be used in the corresponding

Givens rotation.

INDXP (workspace) INTEGER array, dimension (N)

The permutation used to place deflated values of D

at the end of the array. INDXP(1:K) points to the nondeflated D

values
and INDXP(K+1:N) points to the deflated eigenval

ues.

INDX (workspace) INTEGER array, dimension (N)

The permutation used to sort the contents of D into

ascending order.

INFO (output) INTEGER

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

gal value.

FURTHER DETAILS

Based on contributions by

Jeff Rutter, Computer Science Division, University of

California
at Berkeley, USA

LAPACK version 3.0 15 June 2000

SLAED8(3)