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claic1

 NAME
      CLAIC1 - apply 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.