A computational comparison of the first nine members of a determinantal family of root-finding methods

摘要

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, B_m~(k) defined as the ratio of two determinants that depend on the first m - k derivatives of the given function. This infinite family is derived in Kalantari (J. Comput. Appl. Math. 126 (2000) 287-318) and its order of convergence is analyzed in Kalantari (BIT 39 (1999) 96-109). In this paper we give a computational study of the first nine root-finding methods. These include Newton, secant, and Halley methods. Our computational results with polynomials of degree up to 30 reveal that for small degree polynomials B_m~(k-1) is more efficient than B_m~(k), but as the degree increases, B_m~(k) becomes more efficient than B_m~(k-1), The most efficient of the nine methods is B_4~(4), having theoretical order of convergence equal to 1.927. Newton's method which is often viewed as the method of choice is in fact the least efficient method.