Fixed Point Iteration Method

We now formally write the algorithm down. If we wish to find the solution for a set of n algebraic equations in n variables:

We first write the equations in the form (there is no unique way to to this!):

We then guess the initial values of all n variables and iterate as follows:

Unfortunately, this method does not converge always to the true solution for all nonlinear equations (as seen in the previous slide) and even if it does it sometimes converges very slowly. Some variations are proposed to improve convergence. For example, for the solution of , instead of using , one may first compute:

where a is a value which is between 0 and 1. It is found that this may speed up convergence for certain nonlinear equations.

Alternative methods like Newton Raphson(N-R) can also be used for solving nonlinear equations. For an equation , N-R algorithm is :

where the partial derivative is evaluated at x=xk.