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 = 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' not equal 0, for each m = 2,...,n + 1, we describe a determinantal interpolation formula, f = P-m,P-n + R-m,R-n where P-m,P-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 Pade approximant. By applying the formulas to P-n, for each m greater than or equal to2, Pm.m-1,...,P-m,P-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 less than or equal tom, 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 pi ,e, and other special numbers. (C) 2000 Elsevier Science B.V. All rights reserved. MSG: 65H05; 65D05; 65Y20; 41A20; 41A21; 30C15; 30E10; 12D10.

[发布日期] 2000-12-30

[关键词] Taylor's theorem;Newton's method;interpolation;confluent divided differences;fixed-point iteration;rootfinding;Halley's method;Pade approximants