Order of Convergence

• The fixed-point method converges at a linear rate;
• it is said to be linearly convergent, meaning that the error at each successive iteration is a constant fraction of the previous error.

Table 1: The order of convergence for the iteration algorithm with the different forms of $x=g(x)$ .

• If we tabulate the errors after each step in getting the roots of the polynomial and its ratio to the previous error,
• we find that the magnitudes of the ratios to be levelling out at 0.3333. (See Table 1)

• Example: Comparing Muller's and Fixed-point Iteration methods (http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter1/mainmulfix.m m-files: mainmulfix.m, http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter1/muller.m muller.m, http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter1/fixedpoint.m fixedpoint.m)