Optimal Iterative Processes for Rootfinding

16. R. P. Brent, S. Winograd and P. Wolfe, Optimal iterative processes for rootfinding, Numerische Mathematik 20 (1973), 327-341. CR 15#26753, MR 47#6079.

Abstract and errata: dvi (2K), pdf (30K), ps (33K).

Paper: pdf (1262K).

Errata: pdf (35K).


Let f0(x) be a function of one variable with a simple zero at r0. An iteration scheme is said to be locally convergent if, for some initial approximation x1, ... , xk near r0 and all functions f which are sufficiently close (in a certain sense) to f0, the scheme generates a sequence (xk) which lies near r0 and converges to a zero r of f. The order of convergence of the scheme is the infimum of the order of convergence of (xk) for all such functions f. We study iteration schemes which are locally convergent and use only evaluations of f, f', ... , fd at x1, ... , xk-1 to determine xk, and we show that no such scheme has order greater than d+2. This bound is the best possible, for it is attained by certain schemes based on polynomial interpolation.

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