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Optimal Order of One-Point and Multipoint Iteration

Authors:

H. T. Kung,

J. F. TraubAuthors Info & Claims

Journal of the ACM (JACM), Volume 21, Issue 4

Pages 643 - 651

https://doi.org/10.1145/321850.321860

Published: 01 October 1974 Publication History

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Abstract

The problem is to calculate a simple zero of a nonlinear function ƒ by iteration. There is exhibited a family of iterations of order 2^n-1 which use n evaluations of ƒ and no derivative evaluations, as well as a second family of iterations of order 2^n-1 based on n — 1 evaluations of ƒ and one of ƒ′. In particular, with four evaluations an iteration of eighth order is constructed. The best previous result for four evaluations was fifth order.

It is proved that the optimal order of one general class of multipoint iterations is 2^n-1 and that an upper bound on the order of a multipoint iteration based on n evaluations of ƒ (no derivatives) is 2ⁿ.

It is conjectured that a multipoint iteration without memory based on n evaluations has optimal order 2^n-1.

References

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KUNG, H. T., AND TRXUB, J.F. Optimal order for iterations using two evaluations. Report, Computer Sci. Dep., Carnegie-Mellon U., Pittsburgh, Pa., 1973. To appear in SIAM J. Numev. Anal.

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Index Terms

Optimal Order of One-Point and Multipoint Iteration
1. Mathematics of computing
  1. Mathematical analysis
    1. Nonlinear equations
    2. Numerical analysis
      1. Number-theoretic computations

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Published: 01 October 1974

Published in JACM Volume 21, Issue 4

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Abstract

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A class of three-point root-solvers of optimal order of convergence

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