DLAED7 - compute the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix

SUBROUTINE DLAED7(

ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK, INFO )

INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N, QSIZ, TLVLS

DOUBLE PRECISION RHO

INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )

DOUBLE PRECISION D( * ), GIVNUM( 2, * ), Q( LDQ, * ), QSTORE( * ), WORK( * )

DLAED7 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 optionally eigenvectors of a dense symmetric matrix that has been reduced to tridiagonal form. DLAED1 handles the case in which all eigenvalues and eigenvectors of a symmetric tridiagonal matrix 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 DLAED8.

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 DLAED9). 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.

ICOMPQ (input) INTEGER

= 0: Compute eigenvalues only.

= 1: Compute eigenvectors of original dense symmetric matrix also. On entry, Q contains the orthogonal matrix used to reduce the original matrix to tridiagonal form.

N (input) INTEGER

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

QSIZ (input) INTEGER

The dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.

TLVLS (input) INTEGER

The total number of merging levels in the overall divide and conquer tree.

CURLVL (input) INTEGER The current level in the overall merge routine, 0 <= CURLVL <= TLVLS.

CURPBM (input) INTEGER The current problem in the current level in the overall merge routine (counting from upper left to lower right).

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 (output) INTEGER array, dimension (N)

The permutation which will reintegrate the subproblem just solved back into sorted order, i.e., D( INDXQ( I = 1, N ) ) will be in ascending order.

RHO (input) DOUBLE PRECISION

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

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

QSTORE (input/output) DOUBLE PRECISION array, dimension (N**2+1) Stores eigenvectors of submatrices encountered during divide and conquer, packed together. QPTR points to beginning of the submatrices.

QPTR (input/output) INTEGER array, dimension (N+2)

List of indices pointing to beginning of submatrices stored in QSTORE. The submatrices are numbered starting at the bottom left of the divide and conquer tree, from left to right and bottom to top.

PRMPTR (input) INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in PERM a level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of the permutation and also the size of the full, non-deflated problem.

PERM (input) INTEGER array, dimension (N lg N)

Contains the permutations (from deflation and sorting) to be applied to each eigenblock.

GIVPTR (input) INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in GIVCOL a level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) indicates the number of Givens rotations.

GIVCOL (input) INTEGER array, dimension (2, N lg N) Each pair of numbers indicates a pair of columns to take place in a Givens rotation.

GIVNUM (input) DOUBLE PRECISION array, dimension (2, N lg N) Each number indicates the S value to be used in the corresponding Givens rotation.

WORK (workspace) DOUBLE PRECISION array, dimension (3*N+QSIZ*N)

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