CXML
        	    
LAPACK version 3.0
zlaic1(3)

PURPOSE
  ZLAIC1 - applie one step of incremental condition estimation in its
  simplest version

SYNTAX
  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 )

DESCRIPTION
  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.

  SESTPR  (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.