A Proof of the Daniel-moore Conjectures for A-stable Multistep Two-derivative Integration Formulas

A simple proof is given of the following particular case of the DanielMoore conjectures, recently proved in full generality by Wanner, Hairer and Nersett. The maximum error order achievable by an A-stable k-step two-derivative formula equals four and the optimum value of the corresponding error constant is achieved by the second Obrechkoff method. The arguments presented in this paper reveal some new properties of twovariable Hurwitz polynomials of degree (k,2) which, when applied to the canonical polynomial associated with the integration formula, directly yield a proof of the Daniel-Moore conjectures in the case considered herein. 1. Introduetion The problem of the maximum error order compatible with A-stability for multistep multiderivative integration formulas has been recently tackled in an important paper by Wanner, Hairer and Nersett 1). In their contribution, in addition to several other results, these authors present a proof of the Daniel-Meere conjectures 2) saying that the maximum error order achievable by an n-derivative formula equals 2n and that the minimum absolute value of the corresponding error constant is (n!)2j(2n)!(2n + I)!. The aim of the present paper is to offer an alternative proof of the DanielMoore conjectures for the case n = 2, which reveals some' interesting properties of two-variable Hurwitz polynomials and of positive algebraic functions of degree 2. These properties have the remarkable feature of being expressible in terms of positive reality of certain one-variable rational functions. In that respect, it is worth mentioning that the power of the approach to stability problems via positive functions is nowadays well recognized, not only in numerical analysis as shown by Dahlquist 3) but also in various other fields of applied mathematics 4-9). *) ThisworkwassupportedinpartbytheU.S.ArmyResearchOfficeunderContractDAAG29-77-C0042 and by the Joint ServicesElectronics Program, under contract DAAG20-79-C-0047, while Y. Genin was on leave of absence at Information Systems Laboratory, Stanford University. Philips Juurnal uf Research VIII. 36 NII.2 1981 77 Ph. Delsarte, Y. Genin and Y. Kamp The twofold aspect of the paper is reflected in its organization. In the first part (secs 2 and 3), the Daniel-Moore conjectures are discussed with the help of the canonical polynomial introduced in ref. 10 and a proof of the conjectures is worked out, which relies upon some properties exhibited by positive algebraic functions of degree 2. The properties in question are then proved in the second part (sec. 4) independently of the Daniel-Moore environment and thus are, it is hoped, capable of applications in other areas. The general properties of A-stable k-step n-derivative formulas as described in ref. 10 are briefly recalled in sec. 2. Two technical modifications with respect to ref. 10 are introduced. The first one pertains to the yery definition of the error order, oddly defined in ref. 10 and corrected here to comply with standard practice in the field as suggested by Gear 11), Jeltsch 12) and Wanner et al. 1). The second modification affects the question of the exact properties of the even and odd parts of the canonical polynomial that are implied by the assumption of asymptotic convergence. The main part of the paper is concerned with the particular case n = 2. Sec. 3 contains the preliminary steps of the proof of the Daniel-Moore conjectures. Even and odd canonical polynomials associated with k-step 2derivative formulas of error order 4 are expressed into a well-defined parametric form. Then the problem reduces to discovering the properties of the one-variable parameter polynomials that result from the stability condition. This problem is solved in sec. 4, in its most natural framework. The main result, which directly yields a proof of the Daniel-Moore conjectures for n = 2, asserts that a certain one-variable rational function derived from a two-variable Hurwitz polynomial of degree (k,2) has to be a positive real function. An interesting interpretation of this result in terms of transformations of Hurwitz polynomials is emphasized. 2. Definitions and preliminaries In the present paper we are mainly interested in A -stable strongly convergent integration methods, but we also consider certain weakly convergent methods in the process of argumentation. Let us recall the algebraic characterization of these properties in terms of the so-called canonical polynomial. Let n k L L (-li ai,j hiDiXt_j = 0, t = k, k + 1, ... (1) i=o j=O . be any k-step n-derivative integration formula, and let H(p,q) = ao(p) + qal(P) + ... + a" an(p) be its associated canonical polynomial P), where (2) 78 Philips Juurnul ut' Research Vol.3(, :\n.2 19KI Proof of the Dan iel-Ma are conjectures for A-stable k ~ a.. (~)k-j ai(P) = (P IIL ',J p _ 1