zlaic1(3S)

NAME
     ZLAIC1 - applie 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 )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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.

     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.

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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