dlaed2 (l) - Linux Manuals

dlaed2: merges the two sets of eigenvalues together into a single sorted set

NAME

DLAED2 - merges the two sets of eigenvalues together into a single sorted set

SYNOPSIS

SUBROUTINE DLAED2(
K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W, Q2, INDX, INDXC, INDXP, COLTYP, INFO )

    
INTEGER INFO, K, LDQ, N, N1

    
DOUBLE PRECISION RHO

    
INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ), INDXQ( * )

    
DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ), W( * ), Z( * )

PURPOSE

DLAED2 merges the two sets of eigenvalues together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more eigenvalues are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one.

ARGUMENTS

K (output) INTEGER
The number of non-deflated eigenvalues, and the order of the related secular equation. 0 <= K <=N.
N (input) INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
N1 (input) INTEGER
The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= N1 <= N/2.
D (input/output) DOUBLE PRECISION array, dimension (N)
On entry, D contains the eigenvalues of the two submatrices to be combined. On exit, D contains the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order.
Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
On entry, Q contains the eigenvectors of two submatrices in the two square blocks with corners at (1,1), (N1,N1) and (N1+1, N1+1), (N,N). On exit, Q contains the trailing (N-K) updated eigenvectors (those which were deflated) in its last N-K columns.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
INDXQ (input/output) INTEGER array, dimension (N)
The permutation which separately sorts the two sub-problems in D into ascending order. Note that elements in the second half of this permutation must first have N1 added to their values. Destroyed on exit.
RHO (input/output) DOUBLE PRECISION
On entry, the off-diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined. On exit, RHO has been modified to the value required by DLAED3.
Z (input) DOUBLE PRECISION array, dimension (N)
On entry, Z contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). On exit, the contents of Z have been destroyed by the updating process. DLAMDA (output) DOUBLE PRECISION array, dimension (N) A copy of the first K eigenvalues which will be used by DLAED3 to form the secular equation.
W (output) DOUBLE PRECISION array, dimension (N)
The first k values of the final deflation-altered z-vector which will be passed to DLAED3.
Q2 (output) DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
A copy of the first K eigenvectors which will be used by DLAED3 in a matrix multiply (DGEMM) to solve for the new eigenvectors.
INDX (workspace) INTEGER array, dimension (N)
The permutation used to sort the contents of DLAMDA into ascending order.
INDXC (output) INTEGER array, dimension (N)
The permutation used to arrange the columns of the deflated Q matrix into three groups: the first group contains non-zero elements only at and above N1, the second contains non-zero elements only below N1, and the third is dense.
INDXP (workspace) INTEGER array, dimension (N)
The permutation used to place deflated values of D at the end of the array. INDXP(1:K) points to the nondeflated D-values
and INDXP(K+1:N) points to the deflated eigenvalues. COLTYP (workspace/output) INTEGER array, dimension (N) During execution, a label which will indicate which of the following types a column in the Q2 matrix is:
1 : non-zero in the upper half only;
2 : dense;
3 : non-zero in the lower half only;
4 : deflated. On exit, COLTYP(i) is the number of columns of type i, for i=1 to 4 only.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

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 dlaed2

  • dlaed2 (3)
  • dlaed0 (l) - computes all eigenvalues and corresponding eigenvectors of a symmetric tridiagonal matrix using the divide and conquer method
  • dlaed1 (l) - computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix
  • 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