NAME¶

CLAED7 - computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix

SYNOPSIS¶

SUBROUTINE CLAED7(

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

INTEGER CURLVL, CURPBM, CUTPNT, INFO, LDQ, N, QSIZ, TLVLS REAL RHO INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * ) REAL D( * ), GIVNUM( 2, * ), QSTORE( * ), RWORK( * ) COMPLEX Q( LDQ, * ), WORK( * )

PURPOSE¶

CLAED7 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 or banded Hermitian matrix that has been reduced to tridiagonal form.
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 SLAED2.
The second stage consists of calculating the updated
eigenvalues. This is done by finding the roots of the secular
equation via the routine SLAED4 (as called by SLAED3).
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.

ARGUMENTS¶

N (input) INTEGER

The dimension of the symmetric tridiagonal matrix. N >= 0. CUTPNT (input) INTEGER Contains the location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N.

QSIZ (input) INTEGER

The dimension of the unitary matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N.

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) REAL array, dimension (N)

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

Q (input/output) COMPLEX 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).

RHO (input) REAL

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

INDXQ (output) INTEGER array, dimension (N)

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

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

RWORK (workspace) REAL array,

dimension (3*N+2*QSIZ*N)

WORK (workspace) COMPLEX array, dimension (QSIZ*N)

QSTORE (input/output) REAL 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) REAL array, dimension (2, N lg N) Each number indicates the S value to be used in the corresponding Givens rotation.

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