SLAED3(l)								 )								 SLAED3(l)

NAME

       SLAED3 - find the roots of the secular equation, as defined by the values in D, W, and RHO, between 1 and K

SYNOPSIS

       SUBROUTINE SLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX, CTOT, W, S, INFO )

	   INTEGER	  INFO, K, LDQ, N, N1

	   REAL 	  RHO

	   INTEGER	  CTOT( * ), INDX( * )

	   REAL 	  D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ), S( * ), W( * )

PURPOSE

       SLAED3  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 SLAED4 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 we know of none.

ARGUMENTS

       K       (input) INTEGER
	       The number of terms in the rational function to be solved by SLAED4.  K >= 0.

       N       (input) INTEGER
	       The number of rows and columns in the Q matrix.	N >= K (deflation may result in N>K).

       N1      (input) INTEGER
	       The location of the last eigenvalue in the leading submatrix.  min(1,N) <= N1 <= N/2.

       D       (output) REAL array, dimension (N)
	       D(I) contains the updated eigenvalues for 1 <= I <= K.

       Q       (output) REAL array, dimension (LDQ,N)
	       Initially the first K columns are used as workspace.  On output the columns 1 to K contain the updated eigenvectors.

       LDQ     (input) INTEGER
	       The leading dimension of the array Q.  LDQ >= max(1,N).

       RHO     (input) REAL
	       The value of the parameter in the rank one update equation.  RHO >= 0 required.

       DLAMDA  (input/output) REAL array, dimension (K)
	       The  first  K  elements	of  this array contain the old roots of the deflated updating problem.	These are the poles of the secular
	       equation. 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.

       Q2      (input) REAL array, dimension (LDQ2, N)
	       The first K columns of this matrix contain the non-deflated eigenvectors for the split problem.

       INDX    (input) INTEGER array, dimension (N)
	       The  permutation used to arrange the columns of the deflated Q matrix into three groups (see SLAED2).  The rows of the eigenvectors
	       found by SLAED4 must be likewise permuted before the matrix multiply can take place.

       CTOT    (input) 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.

       W       (input/output) REAL array, dimension (K)
	       The first K elements of this array contain the components of the deflation-adjusted updating vector. Destroyed on output.

       S       (workspace) REAL 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.

       LDS     (input) INTEGER
	       The leading dimension of S.  LDS >= max(1,K).

       INFO    (output) INTEGER
	       = 0:  successful exit.
	       < 0:  if INFO = -i, the i-th argument had an illegal value.
	       > 0:  if INFO = 1, an eigenvalue did not converge

FURTHER DETAILS

       Based on contributions by
	  Jeff Rutter, Computer Science Division, University of California
	  at Berkeley, USA
       Modified by Francoise Tisseur, University of Tennessee.

LAPACK version 3.0						   15 June 2000 							 SLAED3(l)