DLAED3 - find the roots of the secular equation, as defined by the values in D, W, and RHO, between 1 and K

SUBROUTINE DLAED3(

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

INTEGER INFO, K, LDQ, N, N1

DOUBLE PRECISION RHO

INTEGER CTOT( * ), INDX( * )

DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ), S( * ), W( * )

DLAED3 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 DLAED4 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.

K (input) INTEGER

The number of terms in the rational function to be solved by DLAED4. 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) DOUBLE PRECISION array, dimension (N)

D(I) contains the updated eigenvalues for 1 <= I <= K.

Q (output) DOUBLE PRECISION 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) DOUBLE PRECISION

The value of the parameter in the rank one update equation. RHO >= 0 required.

DLAMDA (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DLAED2). The rows of the eigenvectors found by DLAED4 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) DOUBLE PRECISION array, dimension (K)

The first K elements of this array contain the components of the deflation-adjusted updating vector. Destroyed on output.

S (workspace) DOUBLE PRECISION 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

Based on contributions by

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Modified by Francoise Tisseur, University of Tennessee.