00112 ^3e1d5508e9adece8aaadfb64a9c05a

112

McWilliams

14]

sjxy(x)-r(x)s>(x)

Since this ratio equals zero if and only if the numerator equals zero, J[x) is minimized by solving the equation:

[5]

s(x)r'(x) - r{x)s\x) = 0

Now suppose that the series expansions of r(x) and ^(x) are given by

r(x) = aQ + a\x + a2x^ +    + ... and s(x) = bo + b\x + b2x^ + bi>x^ + ...

so that

r'(x) = a\ + 2a2x + 3«3x^ + ... and s'(^x) = b\ + 2b2x + 363*^ + ...    [7]

To find an approximate solution, substitute these expansions into [5], ignore higher-order terms, and gather terms having like powers of x to obtain a quadratic eąuation in x:

(boai -aob[) + (2^0 ~ 2ao^A2)*

+(3cr$bo + 2a2b\ + a\b2 - 3aQb3 - 2a{b2 - a2bl)x^ ~ 0    [8]

This approach is applied to the problem of minimizing the cost function [ 1 ] as a function of h. Series expansions were easier to develop when both the numerator and denominator of (1] were multiplied by Ah, and for mathematical convenience the substitution x = Xh was madę and the optimal value x was calculated (the positive root of Eąuation (8J was used). Finally, the optimal value of h was found according to h = x IX. Series expansion coefficients work out to be: