Write a MATLAB function, called f ixed_point_iteration that inputs a function, g

Question

Write a MATLAB function, called f ixed_point_iteration that inputs a function, g, an initial guess x_0, an error tolerance, tol, and a maximum number of iterations, N, and outputs the fixed point of g, obtained using the fixed point iteration, starting with x_0. Your function should have an error defined by E = |x_n - x_n - 1|, and stop when the error is less than the tolerance, or if the number of iterations exceeds N- whichever happens first. Your function header should look something like: function [c, n, err] = fixed_point_iteration (g, x0, tol, N)

Explanation / Answer

Here is the solution for the same. It has been commented for your reference.

Note, the text following '%' symbol is a comment in matlab

function [c,n,err] = fixed_point_iteration(g,x0,tol,N)
% here g is a function, x0 is an initial guess
% tol is the tolerance level and N is number of iterations
intermediate_value = 0; % we use a variable to store intermediate function values

for i=1:N %loop from 1 to N for N iterations
intermediate_value = g(x0); %evaluate function at x0
if abs(intermediate_value - x0) < tol %check if absolute difference is less than tolerance
break; %if difference less than tolerance, then break.
end
x0 = intermediate_value; %update the guess value, to intermediate value
end

n = i;
err = abs(intermediate_value - x0);
c = intermediate_value;

Useage example:

%define some function some_random_function, which given an x returns some y after some computation

function [y] = some_random_function(x)

y=10/(x*x*x - 1);

%create function handler for that function. Function handlers are created by putting '@' sign infront of function name

g = @some_random_function

[a,b,c] = fixed_point_iteration(g, 2, 0.1, 4)

we get output as:

a =

-10.0035

b =

4

c =

10.0743

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