ZLAED7(l) ) ZLAED7(l)NAME
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, GIV‐
COL, 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 mod‐
ification by a rank-one symmetric matrix. This routine is used only for
the eigenproblem which requires all eigenvalues and optionally eigen‐
vectors 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.
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 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 eigen‐
value 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) 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.
LDQ (input) INTEGER
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 modi‐
fication.
INDXQ (output) INTEGER array, dimension (N)
This contains the permutation which will reintegrate the sub‐
problem 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 submatrices encountered during divide and
conquer, packed together. QPTR points to beginning of the subma‐
trices.
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 correspond‐
ing 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
LAPACK version 3.0 15 June 2000 ZLAED7(l)
