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5. E. C. TITCHMARSH, The Theory of Functions, Oxford Univ. Press, London, 1939. 6. E. C. TITCHMARSH, Eigenfunction Expansions Associated With Second-Order Differential Equations, Oxford, at the Clarendon Press, 1946; Part I, Oxford Univ. Press, London, 1962. MR 8, 458. 7. G. N. WATSON, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press and Macmillan, New York, 1944. MR 6, 64. 8. W. H. YOUNG, "On series of Bessel functions," Proc. London Math. Soc., (2), v. 18, 1920, pp. 163-200.

In this paper we shall give an extension of Ostrowski's point of attraction theorem to multistep iterative procedures for finding a zero of a nonljnear function defined on $R^n $. We shall also obtain a rate of convergence statement in terms of the spectral radius of certain matrices. These results will then be applied to a class of procedures obtained by composing one-dimensional iterative methods with the Jacobi and successive-overrelaxation processes. The rate of convergence of these procedures will be shown to be independent of the rate of convergence of the one-dimensional methods.

This paper is concerned with the order of convergence of iterative procedures for finding a zero of a nonlinear function defined on $R^n $. Some of the results provide conditions for determining the precise order of convergence of a method rather than the usual lower bound on the order. These precise order results are applied to a large class of methods that include many well-known techniques such as the secant method and Steffenson’s method.