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

Abstract

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= P n + R n , where P n is the Newton interpolating polynomial computed with respect to a confluent vector of nodes, and R n is the remainder. Whenever f′≠0, for each m=2,…, n+1, we describe a “determinantal interpolation formula”, f= P m, n + R m, n , where P m, 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 P n , for each m⩾2, P m,m−1,…,P m,m+n−2 is a set of n rational approximations that includes P n , and may provide a better approximation to f, than P n . 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 B m ( 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 B m (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 B m ( 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 B m (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.