# ?laed8

Used by ?stedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

## Syntax

call slaed8( icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z, dlamda, q2, ldq2, w, perm, givptr, givcol, givnum, indxp, indx, info )

call dlaed8( icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z, dlamda, q2, ldq2, w, perm, givptr, givcol, givnum, indxp, indx, info )

call claed8( k, n, qsiz, q, ldq, d, rho, cutpnt, z, dlamda, q2, ldq2, w, indxp, indx, indxq, perm, givptr, givcol, givnum, info )

call zlaed8( k, n, qsiz, q, ldq, d, rho, cutpnt, z, dlamda, q2, ldq2, w, indxp, indx, indxq, perm, givptr, givcol, givnum, info )

## Description

This routine 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.

## Input Parameters

icompq

  INTEGER. Used with real flavors only.

  If `icompq = 0`, compute eigenvalues only.

  If `icompq = 1`, compute eigenvectors of original dense symmetric matrix also.

  On entry, the array q must contain the orthogonal matrix used to reduce the original matrix to tridiagonal form.

n

  INTEGER. The dimension of the symmetric tridiagonal matrix (`n ≥ 0`).

cutpnt

  INTEGER. The location of the last eigenvalue in the leading sub-matrix. `min(1,n) ≤ cutpnt ≤ n` .

qsiz

  INTEGER.

  The dimension of the orthogonal/unitary matrix used to reduce the full matrix to tridiagonal form; `qsiz ≥ n` (for real flavors, `qsiz ≥ n` if `icompq = 1)`.

d, z

  REAL for slaed8/claed8

  DOUBLE PRECISION for dlaed8/zlaed8.

  Arrays, dimension at least max(1, n) each. The array d(*) contains the eigenvalues of the two submatrices to be combined.

  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). The contents of z are destroyed by the updating process.

q

  REAL for slaed8

  DOUBLE PRECISION for dlaed8

  COMPLEX for claed8

  COMPLEX*16 for zlaed8.

  Array

  q(ldq, *). The second dimension of q must be at least `max(1, n)`. On entry, q contains the eigenvectors of the partially solved system which has been previously updated in matrix multiplies with other partially solved eigensystems.

  For real flavors, If `icompq = 0`, q is not referenced.

ldq

  INTEGER. The first dimension of the array q; `ldq ≥ max(1, n)`.

ldq2

  INTEGER. The first dimension of the output array q2; `ldq2 ≥ max(1, n)`.

indxq

  INTEGER. Array, dimension (n).

  The permutation that 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

  REAL for slaed8/claed8

  DOUBLE PRECISION for dlaed8/zlaed8.

  On entry, the off-diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined.

## Output Parameters

k

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

d

  On exit, contains the trailing (n-k) updated eigenvalues (those which were deflated) sorted into increasing order.

z

  On exit, the updating process destroys the contents of z.

q

  On exit, q contains the trailing (n-k) updated eigenvectors (those which were deflated) in its last (n-k) columns.

indxq

  INTEGER. Array, dimension (n).

  The permutation of merged eigenvalues set.

rho

  On exit, rho has been modified to the value required by ?laed3.

dlamda, w

  REAL for slaed8/claed8

  DOUBLE PRECISION for dlaed8/zlaed8.

  Arrays, dimension (n) each. The array dlamda(*) contains a copy of the first k eigenvalues which will be used by ?laed3 to form the secular equation.

  The array w(*) will hold the first k values of the final deflation-altered z-vector and will be passed to ?laed3.

q2

  REAL for slaed8

  DOUBLE PRECISION for dlaed8

  COMPLEX for claed8

  COMPLEX*16 for zlaed8.

  Array

  q2(ldq2, *). The second dimension of q2 must be at least `max(1, n)`.

  Contains a copy of the first k eigenvectors which will be used by slaed7/dlaed7 in a matrix multiply (sgemm/dgemm) to update the new eigenvectors. For real flavors, If `icompq = 0`, q2 is not referenced.

indxp, indx

  INTEGER. Workspace arrays, dimension (n) each.

  The array indxp(*) will contain the permutation used to place deflated values of d at the end of the array. On output, indxp(1:k) points to the nondeflated d-values and indxp(k+1:n) points to the deflated eigenvalues.

  The array indx(*) will contain the permutation used to sort the contents of d into ascending order.

perm

  INTEGER. Array, dimension (n ).

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

givptr

  INTEGER. Contains the number of Givens rotations which took place in this subproblem.

givcol

  INTEGER. Array, dimension (2, n ).

  Each pair of numbers indicates a pair of columns to take place in a Givens rotation.

givnum

  REAL for slaed8/claed8

  DOUBLE PRECISION for dlaed8/zlaed8.

  Array, dimension (2, n).

  Each number indicates the S value to be used in the corresponding Givens rotation.

info

  INTEGER.

  If `info = 0`, the execution is successful.

  If `info = -i`, the i-th parameter had an illegal value.

Parent topic: Auxiliary Routines