NAME¶

SLAED1 - computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix

SYNOPSIS¶

SUBROUTINE SLAED1(

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

INTEGER CUTPNT, INFO, LDQ, N REAL RHO INTEGER INDXQ( * ), IWORK( * ) REAL D( * ), Q( LDQ, * ), WORK( * )

PURPOSE¶

SLAED1 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. SLAED7 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 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.

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) REAL 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) REAL

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) REAL 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

FURTHER DETAILS¶

Based on contributions by
Jeff Rutter, Computer Science Division, University of California
at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee.