Maximum polynomial degree Nordsieck-Gear \((k,p)\) methods: Existence, stability, consistency, refinement, convergence and computational examples (Q861198)

This paper is the third in a sequence of papers by the authors on maximum polynomial degree \((k,p)\) Nordsieck-Gear methods for non-stiff ordinary differential equations. The particular features of this paper are that two conjectures from the earlier work are proved and this leads to proofs of convergence and order for the \((k,1)\) method. Details are given relating to implementation and the paper concludes with computational examples that illustrate the results. [cf. \textit{R. Danchick} and \textit{M. L. Juncosa}, Numerical ordinary differential equations, 1979 SIGNUM Meet., Champaign, Ill., Exp. No.~23, 1--2 (1979; Zbl 0427.65054); ICNAAM 2005, 144--147 (2005; Zbl 1087.65569)]