Some multipoint iterative methods without memory for approximating simple zeros of functions of one variable are described. For m>0, n>O, and m+l>k>0, there exist methods which use one evaluation of f, f', ... , f(m) followed by n evaluations of f(k) for each iteration, and have order of convergence m+2n+l. In particular, there are methods of order 2(n+l) 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. 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. Numerical examples and some Fortran subroutines are given.
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