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NAME
     zlaic1 - apply one step of incremental condition  estimation
     in its simplest version

SYNOPSIS
     SUBROUTINE ZLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )

     INTEGER J, JOB

     DOUBLE PRECISION SEST, SESTPR

     COMPLEX*16 C, GAMMA, S

     COMPLEX*16 W( J ), X( J )



     #include <sunperf.h>

     void zlaic1(int job, int j, doublecomplex *zx, double  sest,
               doublecomplex  *w,  doublecomplex  *gamma,  double
               *sestpr, doublecomplex *s, doublecomplex *c) ;

PURPOSE
     ZLAIC1 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 ZLAIC1 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*16 array, dimension (J)
               The j-vector x.

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

     W         (input) COMPLEX*16 array, dimension (J)
               The j-vector w.

     GAMMA     (input) COMPLEX*16
               The diagonal element gamma.

     SEDTPR    (output) DOUBLE PRECISION
               Estimated singular value of (j+1) by (j+1)  matrix
               Lhat.

     S         (output) COMPLEX*16
               Sine needed in forming xhat.

     C         (output) COMPLEX*16
               Cosine needed in forming xhat.

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