ZLAED7 - compute the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix
SYNOPSIS
SUBROUTINE ZLAED7( 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
DOUBLE PRECISION RHO
INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( *
), IWORK( * ), PERM( * ), PRMPTR( * ),
QPTR( * )
DOUBLE PRECISION D( * ), GIVNUM( 2, * ),
QSTORE( * ), RWORK( * )
COMPLEX*16 Q( LDQ, * ), WORK( * )
PURPOSE
ZLAED7 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 DLAED2.
eigenvalues. This is done by finding the roots of
the secular
equation via the routine DLAED4 (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 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.
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 (count
ing 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) COMPLEX*16 array, dimension (LDQ,N)
On entry, the eigenvectors of the rank-1-perturbed
matrix. On exit, the eigenvectors of the repaired
tridiagonal matrix.
The leading dimension of the array Q. LDQ >=
max(1,N).
RHO (input) DOUBLE PRECISION
Contains the subdiagonal element used to create the
rank-1 modification.
INDXQ (output) INTEGER array, dimension (N)
This contains the permutation which will reinte
grate 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) DOUBLE PRECISION array,
dimension (3*N+2*QSIZ*N)
WORK (workspace) COMPLEX*16 array, dimension (QSIZ*N)
QSTORE (input/output) DOUBLE PRECISION array,
dimension (N**2+1) Stores eigenvectors of submatri
ces encountered during divide and conquer, packed
together. QPTR points to beginning of the submatri
ces.
QPTR (input/output) INTEGER array, dimension (N+2)
List of indices pointing to beginning of submatri
ces stored in QSTORE. The submatrices are numbered
starting at the bottom left of the divide and con
quer 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 prob
lem.
PERM (input) INTEGER array, dimension (N lg N)
Contains the permutations (from deflation and sort
ing) 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.
(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 ille
gal value.
> 0: if INFO = 1, an eigenvalue did not converge
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