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

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, Bm(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 Bm(k−1) is more efficient than Bm(k), but as the degree increases, Bm(k) becomes more efficient than Bm(k−1). The most efficient of the nine methods is B4(4), having theoretical order of convergence equal to 1.927. Newtonʹs method which is often viewed as the method of choice is in fact the least efficient method.