CLAIC1(3f)          LAPACK auxiliary routine (version 3.1)          CLAIC1(3f)



NAME
       CLAIC1 - 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 auxiliary routine (versioNovember 2006                      CLAIC1(3f)
