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Generalization of Taylor's theorem and Newton's method via a new family of determinantal interpolation formulas and its applications by Bahman Kalantari is a Mathematics article available to read on EtoBox.

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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 con uent 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 Padà e 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 inÿnite spectrum of rational approximations. The formulas also give an inÿnite spectrum of rational inverse approximations, as well as a fundamental k-point iteration function B (k) m , for each k6m, deÿned as the ratio of two determinants that depend on the ÿrst m -k derivatives. Application of our formulas have motivated several new results obtained in seq

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Author

Bahman Kalantari

Publisher

Elsevier Science; Elsevier ; Elsevier BV (ISSN 0377-0427)

Published

2000

Language

EN

Field

Mathematics (Physical Sciences)