DLAED1 - compute the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix

SUBROUTINE DLAED1(

N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, INFO )

INTEGER CUTPNT, INFO, LDQ, N

DOUBLE PRECISION RHO

INTEGER INDXQ( * ), IWORK( * )

DOUBLE PRECISION D( * ), Q( LDQ, * ), WORK( * )

DLAED1 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. This routine is used only for the eigenproblem which requires all eigenvalues and eigenvectors of a tridiagonal matrix. DLAED7 handles the case in which eigenvalues only or eigenvalues and eigenvectors of a full symmetric matrix (which was reduced to tridiagonal form) are desired.

T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)

where Z = Q'u, u is a vector of length N with ones in the CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.

The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages:

The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLAED2.

The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine DLAED4 (as called by DLAED3). This routine also calculates the eigenvectors of the current problem.

The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.

N (input) INTEGER

The dimension of the symmetric tridiagonal matrix. N >= 0.

D (input/output) DOUBLE PRECISION array, dimension (N)

On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.

Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)

On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.

LDQ (input) INTEGER

The leading dimension of the array Q. LDQ >= max(1,N).

INDXQ (input/output) INTEGER array, dimension (N)

On entry, the permutation which separately sorts the two subproblems in D into ascending order. On exit, the permutation which will reintegrate the subproblems back into sorted order, i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.

RHO (input) DOUBLE PRECISION

The subdiagonal entry used to create the rank-1 modification.

CUTPNT (input) INTEGER The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N/2.

WORK (workspace) DOUBLE PRECISION array, dimension (4*N + N**2)

IWORK (workspace) INTEGER array, dimension (4*N)

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.