H. T. Kung
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 2n-1 which use n evaluations of ƒ and no derivative evaluations, as well as a second family of iterations of order 2n-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 2n-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 2n-1.
- 1 BRENT, R., WINOGRAD, S., AND WOLFE, P. Optimal iterative processes for root-finding. Numer. Math. ~0 (1973), 327-341.Google Scholar
- 2 JARaATT, P. Some efficient fourth order multipoint methods for solving equations. BIT 9 (1969), 119-124.Google Scholar
- 3 KING, R.R. A family of fourth-order methods for nonlinear equations. SIAM J. Numer. Anal. 10 (1973), 876-879.Google Scholar
- 4 KROGH, F.T. Efficient algorithms for polynomial interpolation and numerical differentiation. Math. Comp. 2~ (1970), 185-190.Google Scholar
- 5 KUNG, H. T., ASh TR~UB, J.F. Computational complexity of one-point and multipoint iteration. Report, Computer Sci. Dep., Carnegie-Mellon U., Pittsburgh, Pa. To appear in Complexity of Real Computation, R. Karp, Ed., American Mathematical Society, Providence, R. I.Google Scholar
- 6 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.Google Scholar
- 7 ORTEGA, J. ~/~., AND RHEINBOLDT, W. C. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970. Google Scholar
- 8 OSTROWSKI, A.M. Solution of Equations and Systems of Equations, 2nd ed. Academic Press, New York, 1966.Google Scholar
- 9 TRAUn, J.F. On functional iteration and the calculation of roots. Proc. 16th Nat. ACM Conf., 5A-1 (1961), pp. 1--4. Google Scholar
- 10 TRAUB, J.F. Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs, N. J., 1964.Google Scholar
- 11 TRAUB, J. F. Computational complexity of iterative processes. SIAM J. Comput. 1 (1972), 167-179.Google Scholar
Index Terms
- Optimal Order of One-Point and Multipoint Iteration
Recommendations
A class of three-point root-solvers of optimal order of convergence
The construction of a class of three-point methods for solving nonlinear equations of the eighth order is presented. These methods are developed by combining fourth order methods from the class of optimal two-point methods and a modified Newton's method ...
A proximal point algorithm for the monotone second-order cone complementarity problem
This paper is devoted to the study of the proximal point algorithm for solving monotone second-order cone complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original ...
A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings
In the present paper, we define and study a new three-step iterative scheme inspired by Suantai [J. Math. Anal. Appl. 311 (2005) 506-517]. This scheme includes many well-known iterations, for examples, modified Mann-type, modified Ishikawa-type ...
Comments