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Optimal Order and Efficiency for Iterations with Two Evaluations

Kung, H.T.; Traub, Josephe F.

The problem is to calculate a simple zero of a nonlinear function f. We consider rational iterations without memory which use two evaluations of f or its derivatives. It is shown that the optimal order is 2. This settles a conjecture of Kung and Traub that an iteration using n evaluations without memory is of order at most 2ⁿ⁻¹, for the case n=2. Furthermore we show that any rational two-evaluation iteration of optimal order must use either two evaluations of f or one evaluation of f and one of f'. From this result we completely settle the question of the optimal efficiency, in our efficiency measure, for any two-evaluation iteration without memory. Depending on the relative cost of evaluating f and f', the optimal efficiency is achieved by either Newton iteration or the iteration ᴪ.

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Also Published In

Title
SIAM Journal on Numerical Analysis
DOI
https://doi.org/10.1137/0713010

More About This Work

Academic Units
Computer Science
Published Here
October 10, 2013