Two-parameter, arbitrary order, exponential approximations for stiff equations

Abstract

Abstract.A two-parameter family of approximations to the exponential function is considered. Constraints on the parameters are determined which guarantee the approximations are ^-acceptable. The suitability of these approximations for 2-point vl-stable exponential fitting is established. Several numerical methods, which produce these approximations when solving y = \v, are presented.

Two-Parameter, Arbitrary Order, Exponential Approximations for Stiff Equations

“…Thus, Pn 2(z; u1; u2) represents a family of approximations to the exponential of arbitrarily high order. This family of approximations is the natural generalization of the families previously considered by Ehle [7]. It is easily verified that several of the exponential approximations of Watts [14] are given by appropriate choice of the parameters.…”

“…Constraints for A -Acceptability of Pn 2(z; pt, p2). To establish the Aacceptability of P" 2, we shall use the same strategy employed in [5] and [7]. Thus, we determine when Pn 2 is bounded by 1 on the imaginary axis and also as Re(-z) -►°°.…”

Composite exponential approximations

“…Its properties are given in Theorem 9, which depends on the following lemma. We continue as in [5]; by deducing a contradiction from the assumption that a* in (0, 1) and z*, Re z* < 0, exist so that Q(z*; d*) = 0. Because Q(z; 1) = Qnn(z) ¥*■ 0 and Q(z; 0) = Qn-x"(z) ¥= 0 for every z in the left half-plane, the assumption and the root locus property [12] imply the existence of real numbers t and â, where á is in (0, 1), such that Q(it; a) = 0.…”

“…Because Q"-h"(-it) and QnJ,-it) are equal to P^-iity and P""iil), respectively, and because the equations Pn,n-i(x) = Pn,n(x) ~ 2(2« -i)x/>"-'"-'(^' Qn-\,n(X) = Qn,ÁX) + 2(2^rT) XQn-l,n-l(X)> are given in [5], it follows that the identity Qn,n(it)Pn-l,n-l(it) + Pn,n(it)Qn-l,n-l(it) = 0 is obtained, which is the same as the equation…”

“…The polynomial Re Q"t"(it)P"-i:"-i(it) is even and, according to [5], the even- (ii) The order of R is proved by combining Theorem 3 and Lemma 6, using the identity Qnm(0) = 1.…”