slaed2(3)

NAME

SLAED2 - merge the two sets of eigenvalues together into a

single sorted set

SYNOPSIS

SUBROUTINE  SLAED2(  K,  N,  N1, D, Q, LDQ, INDXQ, RHO, Z,
DLAMDA, W, Q2, INDX, INDXC, INDXP, COLTYP, INFO )
    INTEGER        INFO, K, LDQ, N, N1
    REAL           RHO
    INTEGER        COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ), INDXQ( * )
    REAL           D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( *
), W( * ), Z( * )

PURPOSE

SLAED2 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 en

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

related secular equation problem is reduced by one.

ARGUMENTS

K (output) INTEGER

The number of non-deflated eigenvalues, and the or

der of the related secular equation. 0 <= K <=N.

N (input) INTEGER

The dimension of the symmetric tridiagonal matrix.

N >= 0.

N1 (input) INTEGER

The location of the last eigenvalue in the leading

sub-matrix. min(1,N) <= N1 <= N/2.

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

On entry, D contains the eigenvalues of the two

submatrices to be combined. On exit, D contains the trailing (N

K) updated eigenvalues (those which were deflated) sorted into

increasing order.

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

On entry, Q contains the eigenvectors of two subma

trices in the two square blocks with corners at (1,1), (N1,N1)

and (N1+1, N1+1), (N,N). 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/output) 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 N1 added to their

values. Destroyed on exit.

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.

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 have

been 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.

W (output) REAL array, dimension (N)

The first k values of the final deflation-altered

z-vector which will be passed to SLAED3.

Q2 (output) REAL array, dimension (N1**2+(N-N1)**2)

A copy of the first K eigenvectors which will be

used by SLAED3 in a matrix multiply (SGEMM) to solve for the new

eigenvectors.

INDX (workspace) INTEGER array, dimension (N)

The permutation used to sort the contents of DLAMDA

into ascending order.

INDXC (output) INTEGER array, dimension (N)

The permutation used to arrange the columns of the

deflated Q matrix into three groups: the first group contains

non-zero elements only at and above N1, the second contains non

zero elements only below N1, and the third is dense.

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.

COLTYP (workspace/output) INTEGER array, dimension

(N) During execution, a label which will indicate which of the

following types a column in the Q2 matrix is:
1 : non-zero in the upper half only;
2 : dense;
3 : non-zero in the lower half only;
4 : deflated. On exit, COLTYP(i) is the number of

columns of type i, for i=1 to 4 only.

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

Modified by Francoise Tisseur, University of Tennessee.

LAPACK version 3.0 15 June 2000

SLAED2(3)