dlaed1 (l) - Linux Manuals

dlaed1: computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix

NAME

DLAED1 - computes 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*Zaq ) Qaq(in) = Q(out) * D(out) * Qaq(out)
where Z = Qaqu, 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 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) 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)
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.

Pages related to dlaed1

  • dlaed1 (3)
  • dlaed0 (l) - computes all eigenvalues and corresponding eigenvectors of a symmetric tridiagonal matrix using the divide and conquer method
  • dlaed2 (l) - merges the two sets of eigenvalues together into a single sorted set
  • dlaed3 (l) - finds the roots of the secular equation, as defined by the values in D, W, and RHO, between 1 and K
  • dlaed4 (l) - subroutine compute the I-th updated eigenvalue of a symmetric rank-one modification to a diagonal matrix whose elements are given in the array d, and that D(i) < D(j) for i < j and that RHO > 0
  • dlaed5 (l) - subroutine compute the I-th eigenvalue of a symmetric rank-one modification of a 2-by-2 diagonal matrix diag( D ) + RHO The diagonal elements in the array D are assumed to satisfy D(i) < D(j) for i < j
  • dlaed6 (l) - computes the positive or negative root (closest to the origin) of z(1) z(2) z(3) f(x) = rho + --------- + ---------- + --------- d(1)-x d(2)-x d(3)-x It is assumed that if ORGATI = .true
  • dlaed7 (l) - computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix
  • dlaed8 (l) - merges the two sets of eigenvalues together into a single sorted set
  • dlaed9 (l) - finds the roots of the secular equation, as defined by the values in D, Z, and RHO, between KSTART and KSTOP
  • dlaeda (l) - computes the Z vector corresponding to the merge step in the CURLVLth step of the merge process with TLVLS steps for the CURPBMth problem