NAME

CLAIC1 - applie 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 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 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 version 2.0  —  08 October 1994