zlaed8(3)

NAME

ZLAED8 - merge the two sets of eigenvalues together into a

single sorted set

SYNOPSIS

SUBROUTINE  ZLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z,
DLAMDA, Q2, LDQ2, W, INDXP, INDX, INDXQ,  PERM,  GIVPTR,  GIVCOL,
GIVNUM, INFO )
    INTEGER         CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N,
QSIZ
    DOUBLE         PRECISION RHO
    INTEGER        GIVCOL( 2, * ), INDX( * ), INDXP( *  ),
INDXQ( * ), PERM( * )
    DOUBLE          PRECISION D( * ), DLAMDA( * ), GIVNUM(
2, * ), W( * ), Z( * )
    COMPLEX*16     Q( LDQ, * ), Q2( LDQ2, * )

PURPOSE

ZLAED8 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

K (output) INTEGER

Contains the number of non-deflated eigenvalues.

This is the order of the related secular equation.

N (input) INTEGER

The dimension of the symmetric tridiagonal matrix.

N >= 0.

QSIZ (input) INTEGER

The dimension of the unitary matrix used to reduce

the dense or band matrix to tridiagonal form. QSIZ >= N if ICOM

PQ = 1.

Q (input/output) COMPLEX*16 array, dimension (LDQ,N)

On entry, Q contains the eigenvectors of the par

tially 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 ).

D (input/output) DOUBLE PRECISION 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.

RHO (input/output) DOUBLE PRECISION

Contains the off diagonal element associated with

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

are now being recombined. RHO is modified during the computation

to the value required by DLAED3.

CUTPNT (input) INTEGER Contains the location of the

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

N.

Z (input) DOUBLE PRECISION array, dimension (N)

On input this vector contains the updating vector

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

row of the second sub-eigenvector matrix). The contents of Z are

destroyed during the updating process.

DLAMDA (output) DOUBLE PRECISION array, dimension

(N) Contains a copy of the first K eigenvalues which will be used

by DLAED3 to form the secular equation.

Q2 (output) COMPLEX*16 array, dimension (LDQ2,N)

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

Contains a copy of the first K eigenvectors which will be used by

DLAED7 in a matrix multiply (DGEMM) to update the new eigenvec

tors.

LDQ2 (input) INTEGER

The leading dimension of the array Q2. LDQ2 >=

max( 1, N ).

W (output) DOUBLE PRECISION array, dimension (N)

This will hold the first k values of the final de

flation-altered z-vector and will be passed to DLAED3.

INDXP (workspace) INTEGER array, dimension (N)

This will contain the permutation used to place de

flated values of D at the end of the array. On output INDXP(1:K)
points to the nondeflated D-values and INDXP(K+1:N)

points to the deflated eigenvalues.

INDX (workspace) INTEGER array, dimension (N)

This will contain the permutation used to sort the

contents of D into ascending order.

INDXQ (input) INTEGER array, dimension (N)

This contains 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.

PERM (output) INTEGER array, dimension (N)

Contains the permutations (from deflation and sort

ing) to be applied to each eigenblock.

GIVPTR (output) INTEGER Contains the number of

Givens rotations 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) DOUBLE PRECISION array, dimension

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

sponding Givens rotation.

INFO (output) INTEGER

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

gal value.

LAPACK version 3.0 15 June 2000

ZLAED8(3)