SIMPLE FIXED•POINT ITERATION

As mentioned above, open methods employ a formula to predict the root. Such a formula can be developed for simple fixed-poil1t iteration (or, as it is also called, one-point iteration or successive substitution) by rearranging the function f(x) = 0 so that x is or side of the equation: x=g(x) This transformation can be accomplished either by algebraic manipulation or by simply adding x to both sides of the original equation.

For example, x^2-2x+3=0 Can be simply manipulated to yield .

 x=(x^2+3)/2

 Whereas sin x=0 could be put into the form of equation by adding x to both sides to yield X=sin x +x The utility of Equation is that it provides a formula to predict a new value of x as a function of an old value of x.

 Thus, given an initial guess at the root Xi, can be used to compute a new estimate Xi+l as expressed by the iterative formula x_(i+1)=g(x_i) As with other iterative formulas in this book, the approximate error for this equation can be determined using the error estimator :

 ε=(x_(i+1)-x_i)/x_(i+1)