Abstract
Recently, we have shown that for each natural number m greater than one, and each natural number k less than or equal to m, there exists a root-finding iteration function, defined as the ratio of two determinants that depend on the first m - k derivatives of the given function. For each k the corresponding matrices are upper Hessenberg matrices. Additionally, for k = 1 these matrices are Toeplitz matrices. The goal of this paper is to analyze the order of convergence of this fundamental family. Newton's method, Halley's method, and their multi-point versions are members of this family. In this paper we also derive these special cases. We prove that for fixed m, as k increases, the order of convergence decreases from m to the positive root of the characteristic polynomial of generalized Fibonacci numbers of order m. For fixed k, the order of convergence increases in m. The asymptotic error constant is also derived in terms of special determinants.
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Kalantari, B. On the Order of Convergence of a Determinantal Family of Root-Finding Methods. BIT Numerical Mathematics 39, 96–109 (1999). https://doi.org/10.1023/A:1022321325108
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DOI: https://doi.org/10.1023/A:1022321325108