DLAED7(1) LAPACK routine (version 3.2) DLAED7(1)NAME
DLAED7 - computes the updated eigensystem of a diagonal matrix after
modification by a rank-one symmetric matrix
SYNOPSIS
SUBROUTINE DLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, LDQ,
INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, PERM,
GIVPTR, GIVCOL, GIVNUM, WORK, IWORK, INFO )
INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N, QSIZ,
TLVLS
DOUBLE PRECISION RHO
INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), IWORK( * ),
PERM( * ), PRMPTR( * ), QPTR( * )
DOUBLE PRECISION D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
QSTORE( * ), WORK( * )
PURPOSE
DLAED7 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 symmetric matrix that has been reduced to tridiago‐
nal form. DLAED1 handles the case in which all eigenvalues and eigen‐
vectors of a symmetric tridiagonal matrix 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 DLAED8.
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 DLAED9).
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
ICOMPQ (input) INTEGER
= 0: Compute eigenvalues only.
= 1: Compute eigenvectors of original dense symmetric matrix
also. On entry, Q contains the orthogonal matrix used to
reduce the original matrix to tridiagonal form.
N (input) INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
QSIZ (input) INTEGER
The dimension of the orthogonal matrix used to reduce the full
matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
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) 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 (output) INTEGER array, dimension (N)
The permutation which will reintegrate the subproblem just
solved back into sorted order, i.e., D( INDXQ( I = 1, N ) ) will
be in ascending order.
RHO (input) DOUBLE PRECISION
The subdiagonal element used to create the rank-1 modification.
CUTPNT (input) INTEGER Contains the location of the last eigen‐
value in the leading sub-matrix. min(1,N) <= CUTPNT <= 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 permu‐
tation 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 rota‐
tions. 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, dimen‐
sion (2, N lg N) Each number indicates the S value to be used in
the corresponding Givens rotation.
WORK (workspace) DOUBLE PRECISION array, dimension (3*N+QSIZ*N)
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
LAPACK routine (version 3.2) November 2008 DLAED7(1)
