Generalization of Taylor's Theorem and Newton's Method via a New Family of Determinantal Interpolation Formulas

The general form of Taylor's theorem gives the formula, f = P n + R n , where P n is the New-ton's interpolating polynomial, computed with respect to a connuent vector of nodes, and R n is the remainder. When f 0 6 = 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 it gives Halley's iteration function, as well as a Pad e ap-proximant. of n rational approximations that includes P n , and may provide a better approximation to f, than P n. Thus each Taylor polynomial unfolds into an innnite spectrum of rational approximations. The formulas also give an innnite spectrum of rational inverse approximations, as well as a family of iteration functions for real or complex root nding, more fundamental than the Euler-Schrr oder family, or any other family. Given m 2, for each k m, we obtain a k-point iteration function, deened as the ratio of two determinants that depend on the rst m ? k derivatives, and Toeplitz for k = 1. The order of convergence ranges from m to the limiting ratio of the generalized Fibonacci numbers of order m. By applying these formulas, Hadamard's inequality, Gerschgorin's theorem, and a new lower bound on determinants , we express roots of numbers, e, and , as the limiting ratio of Toeplitz determinants.