DLAIC1(3)      LAPACK routine of NEC Numeric Library Collection      DLAIC1(3)



NAME
       DLAIC1

SYNOPSIS
       SUBROUTINE DLAIC1 (JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)



PURPOSE
            DLAIC1 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 DLAIC1 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 DOUBLE PRECISION array, dimension (J)
                     The j-vector x.

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

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

           GAMMA     (input)
                     GAMMA is DOUBLE PRECISION
                     The diagonal element gamma.

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

           S         (output)
                     S is DOUBLE PRECISION
                     Sine needed in forming xhat.

           C         (output)
                     C is DOUBLE PRECISION
                     Cosine needed in forming xhat.



LAPACK routine                  31 October 2017                      DLAIC1(3)