dlaed3 man page on Scientific

```DLAED3(1)		 LAPACK routine (version 3.2)		     DLAED3(1)

NAME
DLAED3  -  finds	 the  roots of the secular equation, as defined by the
values in D, W, and RHO, between 1 and K

SYNOPSIS
SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX, CTOT, W,
S, INFO )

INTEGER	  INFO, K, LDQ, N, N1

DOUBLE	  PRECISION RHO

INTEGER	  CTOT( * ), INDX( * )

DOUBLE	  PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
S( * ), W( * )

PURPOSE
DLAED3 finds the roots of the secular equation, as defined by the  val‐
ues  in D, W, and RHO, between 1 and K.	It makes the appropriate calls
to DLAED4 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
DLAED4.	K >= 0.

N       (input) INTEGER
The number of rows and columns in the Q matrix.	N >= K (defla‐
tion 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) DOUBLE PRECISION array, dimension (N)
D(I) contains the updated eigenvalues for 1 <= I <= K.

Q       (output) DOUBLE PRECISION 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) DOUBLE PRECISION
The  value  of  the  parameter in the rank one update equation.
RHO >= 0 required.

DLAMDA  (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DLAED2).  The rows of the	eigen‐
vectors	found  by  DLAED4 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) DOUBLE PRECISION array, dimension (K)
The first K elements of this array contain  the	components  of
the deflation-adjusted updating vector. Destroyed on output.

S       (workspace) DOUBLE PRECISION 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 routine (version 3.2)	 November 2008			     DLAED3(1)
```
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