Generalization of Taylor's theorem and Newton's method via a new family of determinantal interpolation formulas and its applications

摘要

The general form of Taylor's theorem for a function f:K→K, where K is the real line or the complex plane, gives the formula, f=Pn+Rn, where Pn is the Newton interpolating polynomial computed with respect to a confluent vector of nodes, and Rn is the remainder. Whenever f′≠0, for each m=2,…,n+1, we describe a “determinantal interpolation formula”, f=Pm,n+Rm,n, where Pm,n is a rational function in x and f itself. These formulas play a dual role in the approximation of f or its inverse. For m=2, the formula is Taylor's and for m=3 is a new expansion formula and a Padé approximant. By applying the formulas to Pn, for each m⩾2,Pm,m−1,…,Pm,m+n−2 is a set of n rational approximations that includes Pn, and may provide a better approximation to f, than Pn. Hence each Taylor polynomial unfolds into an infinite spectrum of rational approximations. The formulas also give an infinite spectrum of rational inverse approximations, as well as a fundamental k-point iteration function Bm(k), for each k⩽m, defined as the ratio of two determinants that depend on the first m−k derivatives. Application of our formulas have motivated several new results obtained in sequel papers: (i) theoretical analysis of the order of Bm(k),k=1,…,m, proving that it ranges from m to the limiting ratio of generalized Fibonacci numbers of order m; (ii) computational results with the first few members of Bm(k) indicating that they outperform traditional root finding methods, e.g., Newton's; (iii) a novel polynomial rootfinding method requiring only a single input and the evaluation of the sequence of iteration functions Bm(1) at that input. This amounts to the evaluation of special Toeplitz determinants that are also computable via a recursive formula; (iv) a new strategy for general root finding; (v) new formulas for approximation of π,e, and other special numbers.