DLAED1 - compute the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix
SYNOPSIS
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( *
)
PURPOSE
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
problem.
The final stage consists of computing the updated
eigenvectors
directly using the updated eigenvalues. The eigen
vectors for
the current problem are multiplied with the eigen
vectors from
the overall problem.
ARGUMENTS
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)
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an ille
gal 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.
Man(1) output converted with
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