Absolute monotonicity of rational functions occurring in the numerical solution of initial value problems

Abstract

Summary. This paper deals with rational functions ~b(z) approximating the exponential function exp(z) related to numerical procedures for solving initial value problems. Motivated by positivity and contractivity requirements imposed on these numerical procedures we study the greatest nonnegative number R, denoted by R(qS), such that ~b is absolutely monotonic on (-R,0]. An algorithm for the computation of R(th) is presented. Application of this algorithm yields the value R(th) for the well-known Pad6 approximations to exp(z). For some specific values of m, n and p we determine the maximum of R(q~) when ~b varies over the class of all rational functions q~ with degree of the numerator <m, degree of the denominator <n and ~b(z)=exp(z)+d~(z p+I) (for z~0).

Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity

“…In this section, we apply Theorem 4.4 in [35] to obtain the radius of absolute monotonicity for linear problems for our SDIRK method. Roughly speaking, this result allows us to restrict the analysis to a finite number of derivatives, that is,…”

“…Proof. In order to apply [35,Theorem 4.4], we construct all the elements (sets, intervals, functions, etc.) involved in this result.…”

“…involved in this result. Following the notation in [35], the stability function (3.15) is of the form R(z) = P (z)/Q(z) with…”

“…More precisely, c(α 0 , 1) ≥ 0, c(α 0 , 2) ≤ 0, and c(α 0 , 3) > 0. Now, we construct the function (see [35,Formula (4.2)])…”

“…Another function needed to construct K(x) in [35] is L(x); in our case, L(x) = 0, see [35,Formula (4.3)]. We can now construct the integer K(x) in [35, Definition 4.1]), defined as the smallest integer k such that k ≥ L(x) and…”