# ?laed3

Used by sstedc/dstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.

## Syntax

call slaed3( k, n, n1, d, q, ldq, rho, dlamda, q2, indx, ctot, w, s, info )

call dlaed3( k, n, n1, d, q, ldq, rho, dlamda, q2, indx, ctot, w, s, info )

## Description

The routine ?laed3 finds the roots of the secular equation, as defined by the values in d, w, and rho, between 1 and k.

It makes the appropriate calls to ?laed4 and then updates the eigenvectors by multiplying the matrix of eigenvectors of the pair of eigensystems being combined by the matrix of eigenvectors of the k-by-k system which is solved here.

This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but none are known.

## Input Parameters

k

  INTEGER. The number of terms in the rational function to be solved by ?laed4 (`k ≥ 0`).

n

  INTEGER. The number of rows and columns in the q matrix. `n ≥ k` (deflation may result in `n >k`).

n1

  INTEGER. The location of the last eigenvalue in the leading sub-matrix; `min(1,n) ≤ n1 ≤ n`/2.

q

  REAL for slaed3

  DOUBLE PRECISION for dlaed3.

  Array `q(ldq, *)`. The second dimension of q must be at least `max(1, n)`.

  Initially, the first k columns of this array are used as workspace.

ldq

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

rho

  REAL for slaed3

  DOUBLE PRECISION for dlaed3.

  The value of the parameter in the rank one update equation. `rho ≥ 0` required.

dlamda, q2, w

  REAL for slaed3

  DOUBLE PRECISION for dlaed3.

  Arrays: `dlamda(k), q2(ldq2, *), w(k)`.

  The first k elements of the array dlamda contain the old roots of the deflated updating problem. These are the poles of the secular equation.

  The first k columns of the array q2 contain the non-deflated eigenvectors for the split problem. The second dimension of q2 must be at least `max(1, n)`.

  The first k elements of the array w contain the components of the deflation-adjusted updating vector.

indx

  INTEGER. Array, dimension (n).

  The permutation used to arrange the columns of the deflated q matrix into three groups (see ?laed2).

  The rows of the eigenvectors found by ?laed4 must be likewise permuted before the matrix multiply can take place.

ctot

  INTEGER. Array, dimension (4).

  A count of the total number of the various types of columns in q, as described in indx. The fourth column type is any column which has been deflated.

s

  REAL for slaed3

  DOUBLE PRECISION for dlaed3.

  Workspace array, dimension (n1+1)*k .

  Will contain the eigenvectors of the repaired matrix which will be multiplied by the previously accumulated eigenvectors to update the system.

## Output Parameters

d

  REAL for slaed3

  DOUBLE PRECISION for dlaed3.

  Array, dimension at least `max(1, n)`.

  d(i) contains the updated eigenvalues for `1 ≤ i ≤ k`.

q

  On exit, the columns 1 to k of q contain the updated eigenvectors.

dlamda

  May be changed on output by having lowest order bit set to zero on Cray X-MP, Cray Y-MP, Cray-2, or Cray C-90, as described above.

w

  Destroyed on exit.

info

  INTEGER.

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

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

  If `info = 1`, an eigenvalue did not converge.

Parent topic: Auxiliary Routines