## Quick Reference

To find a root of an equation *f*(*x*)=0 by the method of fixed-point iteration, the equation is first rewritten in the form *x*=*g*(*x*). Starting with an initial approximation *x*_{0} to the root, the values *x*_{1}, *x*_{2}, *x*_{3},…are calculated using *x*_{n+1}=g(x_{n}). The method is said to converge if these values tend to a limit *α*. If they do, then *α*=*g*(*α*) and so α is a root of the original equation.

A root of *x*=*g*(*x*) occurs where the graph *y*=*g*(*x*) meets the line *y*=*x*. It can be shown that, if |*g*′(*x*)| < 1 in an interval containing both the root and the value *x*_{0}, the method will converge, but not if |*g*′(*x*)|>1. This can be illustrated in figures such as those shown which are for cases in which *g*′(*x*) is positive. For example, the equation *x*^{3}−*x*−1=0 has a root between 1 and 2, so take *x*_{0}=1.5. The equation can be written in the form *x*=*g*(*x*) in several ways, such as (i) *x*=*x*^{3}−1 or (ii) *x*=(*x*+1)^{1/3}. In case (i), *g*′(*x*)=3*x*^{2}, which does not satisfy |*g*′(*x*)| < 1 near *x*_{0}. In case (ii), *g*′(*x*)=⅓(*x*+1)^{−2/3} and *g*′(1.5) ≈ 0.2, so it is likely that with this formulation the method converges.

*Subjects:*
Mathematics.