Starting in maximum-polynomial-degree Nordsieck-Gear methods

摘要

The intent of this paper is to show that the Nordsieck-Gear methods with maximum polynomial degree k+1, first described in [1], admit of matched starting methods which are exact for all polynomials of degree ⩽k+1. In general, it is shown that these starting methods yield starting errors of the required order, O(hk+2), for all initial-value problemsy(P)(x)=f(x,y,y(1),y(2),…,y(p−1)),y(0)=y0, yi(0)=y0(i), i=1,2,…,p−1,where f is k+1 times continuously differentiable in a neighborhood of the graph of the exact solution (x,ȳ(x)), xϵ[0,X]. Two theorems are proved. The first is the constructive existence of an algorithm which requires (k-p+1)(k-p+2)/2 evaluations of the function f to obtain approximations of the method's required higher-order scaled derivatives at the origin:hp+1ȳ(p+1)(0)(p+1)!,…,hkȳ(k)(0)k!,each of which is accurate to O(hk+2). The second, less general theorem, shows that when f is a polynomial in x, y, and its higher order derivatives y(1),y(2),…,y(p−1), an algorithm can be constructed for obtaining the higher-order scaled derivatives exactly. These results lay to rest once and for all any heuristic arguments against varying corrector minus predictor coefficients for preserving maximal order (polynomial degree) because starting values are inexact. Furthermore, and perhaps most importantly, the maximum-polynomial-degree Nordsieck-Gear methods are shown to have a unique property of zero starting error for an important class of ordinary differential equations.