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# slaic1

```
NAME
SLAIC1 - apply one step of incremental condition estimation
in its simplest version

SYNOPSIS
SUBROUTINE SLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )

INTEGER        J, JOB

REAL           C, GAMMA, S, SEST, SESTPR

REAL           W( J ), X( J )

PURPOSE
SLAIC1 applies one step of incremental condition estimation
in its simplest 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 SLAIC1 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] * [ alpha ]
[ gamma ]

where  alpha =  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) REAL array, dimension (J)

The j-vector x.

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

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

GAMMA   (input) REAL
The diagonal element gamma.

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

S       (output) REAL
Sine needed in forming xhat.

C       (output) REAL
Cosine needed in forming xhat.
```