slaic1(3F)

NAME
     SLAIC1 - applie 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.