DLAED2(3f)

NAME
DLAED2 - 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.