claic1 man page on Scientific

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```CLAIC1(1)	    LAPACK auxiliary routine (version 3.2)	     CLAIC1(1)

NAME
CLAIC1  -  applies  one step of incremental condition estimation in its
simplest version

SYNOPSIS
SUBROUTINE CLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )

INTEGER	  J, JOB

REAL		  SEST, SESTPR

COMPLEX	  C, GAMMA, S

COMPLEX	  W( J ), X( J )

PURPOSE
CLAIC1 applies one step of incremental condition estimation in its sim‐
plest version: Let x, twonorm(x) = 1, be an approximate singular vector
of an j-by-j lower triangular matrix L, such that
twonorm(L*x) = sest
Then CLAIC1 computes sestpr, s, c such that
the vector
[ s*x ]
xhat = [  c  ]
is an approximate singular vector of
[ L     0  ]
Lhat = [ w' gamma ]
in the sense that
twonorm(Lhat*xhat) = sestpr.
Depending on JOB, an estimate for  the  largest	or  smallest  singular
value is computed.
Note that [s c]' and sestpr**2 is an eigenpair of the system
diag(sest*sest, 0) + [alpha	gamma] * [ conjg(alpha) ]
[ conjg(gamma) ] where	 alpha
=  conjg(x)'*w.

ARGUMENTS
JOB     (input) INTEGER
= 1: an estimate for the largest singular value is computed.
= 2: an estimate for the smallest singular value is computed.

J       (input) INTEGER
Length of X and W

X       (input) COMPLEX array, dimension (J)
The j-vector x.

SEST    (input) REAL
Estimated singular value of j by j matrix L

W       (input) COMPLEX array, dimension (J)
The j-vector w.

GAMMA   (input) COMPLEX
The diagonal element gamma.

SESTPR  (output) REAL
Estimated singular value of (j+1) by (j+1) matrix Lhat.

S       (output) COMPLEX
Sine needed in forming xhat.

C       (output) COMPLEX
Cosine needed in forming xhat.

LAPACK auxiliary routine (versioNovember 2008			     CLAIC1(1)
```
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