The Secant Method and the Golden Mean

Abstract

The rate at which the sequence {Xk} converges to a root a depends on the multiplicity of a. Recall that a is a root of multiplicity m of the function f if f(x) can be written as f(x) (x a) m(x), where 4 is bounded at a and ( a) = 0. (2) It is well known (see [2]) that if a is simple root, that is, if m = 1, then there will be a nonzero asymptotic error constant C such that the errors Ek q Xk satisfy