Abstract: dvi (2K), pdf (82K).
Paper: pdf (1000K).
In particular, there are methods of order 2(n+1) which use one function evaluation and n+1 derivative evaluations per iteration. These methods naturally generalize the known cases n=0 (Newton's method) and n=1 (Jarratt's fourth-order method), and are useful if derivative evaluations are less expensive than function evaluations.
To establish the order of convergence of the methods we prove some results, which may be of independent interest, on orthogonal and "almost orthogonal" polynomials.
Explicit, nonlinear, Runge-Kutta methods for the solution of a special class of ordinary differential equations may be derived from the methods for finding zeros of functions. The theoretical results are illustrated by several numerical examples.
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