 SLAED3(3) finds the roots of the secular equation, as defined by the values in D, W, and RHO, between 1 and K

## SYNOPSIS

SUBROUTINE SLAED3(
K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX, CTOT, W, S, INFO )

INTEGER INFO, K, LDQ, N, N1

REAL RHO

INTEGER CTOT( * ), INDX( * )

REAL D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ), S( * ), W( * )

## PURPOSE

SLAED3 finds the roots of the secular equation, as defined by the values in D, W, and RHO, between 1 and K. It makes the appropriate calls to SLAED4 and then updates the eigenvectors by multiplying the matrix of eigenvectors of the pair of eigensystems being combined by the matrix of eigenvectors of the K-by-K system which is solved here.
This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

## ARGUMENTS

K (input) INTEGER
The number of terms in the rational function to be solved by SLAED4. K >= 0.
N (input) INTEGER
The number of rows and columns in the Q matrix. N >= K (deflation may result in N>K).
N1 (input) INTEGER
The location of the last eigenvalue in the leading submatrix. min(1,N) <= N1 <= N/2.
D (output) REAL array, dimension (N)
D(I) contains the updated eigenvalues for 1 <= I <= K.
Q (output) REAL array, dimension (LDQ,N)
Initially the first K columns are used as workspace. On output the columns 1 to K contain the updated eigenvectors.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
RHO (input) REAL
The value of the parameter in the rank one update equation. RHO >= 0 required.
DLAMDA (input/output) REAL array, dimension (K)
The first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation. May be changed on output by having lowest order bit set to zero on Cray X-MP, Cray Y-MP, Cray-2, or Cray C-90, as described above.
Q2 (input) REAL array, dimension (LDQ2, N)
The first K columns of this matrix contain the non-deflated eigenvectors for the split problem.
INDX (input) INTEGER array, dimension (N)
The permutation used to arrange the columns of the deflated Q matrix into three groups (see SLAED2). The rows of the eigenvectors found by SLAED4 must be likewise permuted before the matrix multiply can take place.
CTOT (input) INTEGER array, dimension (4)
A count of the total number of the various types of columns in Q, as described in INDX. The fourth column type is any column which has been deflated.
W (input/output) REAL array, dimension (K)
The first K elements of this array contain the components of the deflation-adjusted updating vector. Destroyed on output.
S (workspace) REAL array, dimension (N1 + 1)*K
Will contain the eigenvectors of the repaired matrix which will be multiplied by the previously accumulated eigenvectors to update the system.
LDS (input) INTEGER
The leading dimension of S. LDS >= max(1,K).
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal 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.