Optimal Order and Efficiency for Iterations with Two Evaluations.
CARNEGIE-MELLON UNIV PITTSBURGH PA DEPT OF COMPUTER SCIENCE
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The problem is to calculate a simple zero of a non-linear function f. The authors 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 2 sup n-1, for the case n 2. Furthermore it is shown 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 the authors completely settle the question of the optimal efficiency, in the efficiency measure, for any two-evaluation iteration without memory. Modified author abstract
- Numerical Mathematics