Abstract
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, dened as the ratio of two determinants that depend on the rst m k derivatives of the given function, and for k = 1 are Toeplitz determinants. In this paper we analyze the order of convergence of this fundamental family. For xed m, as k increases, the order decreases from m to the positive root of the characteristic polynomial of generalized Fibonacci numbers of order m. For xed k, the order increases in m. The asymptotic error constant is defined in terms of Toeplitz determinants. Newton's method, Halley's method, and their multipoint versions are members of the family.