NAME

       dlaic1.f -

SYNOPSIS

   Functions/Subroutines
       subroutine dlaic1 (JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)
           DLAIC1 applies one step of incremental condition estimation.

Function/Subroutine Documentation

   subroutine dlaic1 (integer JOB, integer J, double precision, dimension( j ) X, double
       precision SEST, double precision, dimension( j ) W, double precision GAMMA, double
       precision SESTPR, double precision S, double precision C)
       DLAIC1 applies one step of incremental condition estimation.

       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.

       Parameters:
           JOB

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

           J

                     J is INTEGER
                     Length of X and W

           X

                     X is DOUBLE PRECISION array, dimension (J)
                     The j-vector x.

           SEST

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

           W

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

           GAMMA

                     GAMMA is DOUBLE PRECISION
                     The diagonal element gamma.

           SESTPR

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

           S

                     S is DOUBLE PRECISION
                     Sine needed in forming xhat.

           C

                     C is DOUBLE PRECISION
                     Cosine needed in forming xhat.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           September 2012

Author

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