DLAED8 - merge the two sets of eigenvalues together into a single sorted set

SUBROUTINE DLAED8(

ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, GIVCOL, GIVNUM, INDXP, INDX, INFO )

INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N, QSIZ

DOUBLE PRECISION RHO

INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ), INDXQ( * ), PERM( * )

DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ), Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )

DLAED8 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 element in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one.

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.

K (output) INTEGER

The number of non-deflated eigenvalues, and the order of the related secular equation.

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.

D (input/output) DOUBLE PRECISION array, dimension (N)

On entry, the eigenvalues of the two submatrices to be combined. On exit, the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order.

Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)

If ICOMPQ = 0, Q is not referenced. Otherwise, on entry, Q contains the eigenvectors of the partially solved system which has been previously updated in matrix multiplies with other partially solved eigensystems. 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) 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 CUTPNT added to their values in order to be accurate.

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.

CUTPNT (input) INTEGER The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N.

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 are 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.

Q2 (output) DOUBLE PRECISION array, dimension (LDQ2,N)

If ICOMPQ = 0, Q2 is not referenced. Otherwise, a copy of the first K eigenvectors which will be used by DLAED7 in a matrix multiply (DGEMM) to update the new eigenvectors.

LDQ2 (input) INTEGER

The leading dimension of the array Q2. LDQ2 >= max(1,N).

W (output) DOUBLE PRECISION array, dimension (N)

The first k values of the final deflation-altered z-vector and will be passed to DLAED3.

PERM (output) INTEGER array, dimension (N)

The permutations (from deflation and sorting) to be applied to each eigenblock.

GIVPTR (output) INTEGER The number of Givens rotations which took place in this subproblem.

GIVCOL (output) INTEGER array, dimension (2, N) Each pair of numbers indicates a pair of columns to take place in a Givens rotation.

GIVNUM (output) DOUBLE PRECISION array, dimension (2, N) Each number indicates the S value to be used in the corresponding Givens rotation.

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.

INDX (workspace) INTEGER array, dimension (N)

The permutation used to sort the contents of D into ascending order.

INFO (output) INTEGER

= 0: successful exit.

< 0: if INFO = -i, the i-th argument had an illegal value.

Based on contributions by

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA