SLAIC1(3)      LAPACK routine of NEC Numeric Library Collection      SLAIC1(3)



NAME
       SLAIC1

SYNOPSIS
       SUBROUTINE SLAIC1 (JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)



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**T 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]**T and sestpr**2 is an eigenpair of the system

                diag(sest*sest, 0) + [alpha  gamma] * [ alpha ]
                                                      [ gamma ]

            where  alpha =  x**T*w.




ARGUMENTS
           JOB       (input)
                     JOB is INTEGER
                     = 1: an estimate for the largest singular value is computed.
                     = 2: an estimate for the smallest singular value is computed.

           J         (input)
                     J is INTEGER
                     Length of X and W

           X         (input)
                     X is REAL array, dimension (J)
                     The j-vector x.

           SEST      (input)
                     SEST is REAL
                     Estimated singular value of j by j matrix L

           W         (input)
                     W is REAL array, dimension (J)
                     The j-vector w.

           GAMMA     (input)
                     GAMMA is REAL
                     The diagonal element gamma.

           SESTPR    (output)
                     SESTPR is REAL
                     Estimated singular value of (j+1) by (j+1) matrix Lhat.

           S         (output)
                     S is REAL
                     Sine needed in forming xhat.

           C         (output)
                     C is REAL
                     Cosine needed in forming xhat.



LAPACK routine                  31 October 2017                      SLAIC1(3)