Computational divided differencing

摘要

This invention concerns the manipulation and restructuring of programs that compute numerical functions using floating-point numbers, for the purpose of controlling round-off error. In particular, it provides a variety of means for transforming a program that computes a numerical function F(x) into a related program that computes divided differences of F: One such transformation creates a program that computes F[x0, x1], the first divided difference of F(x), where 1 $F\left[x_0,x_1\right]\stackrel{\mathrm{def}}{=}\{\begin{array}{cc}\frac{F\left(x_0\right)-F\left(x_1\right)}{x_0-x_1}& \mathrm{if}{x}_0\ne x_1\\ \frac{d}{dz}F\left(z\right),\mathrm{evaluated}\mathrm{at}z={}_{}\end{array}$