Question

Using Mat lab programming, write the code to solve the given problems by specified methods in each of the following questions. Then discus the simulation results you obtain briefly and prepare the report.

A. Find the roots of the equation, 2x-6X + 3 = 0 by using fixed point iterative method, up to the error criteria, ea = 10⁻⁴. Use X0 = 0 as initial guess

Answer 1

The root of the equation is found after just one iteration and is approximately 3.0000

Fixed Point Iteration for Equation: 2x-6x + 3 = 0

Matlab Code:

Matlab

function root = fixed_point(f, x0, ea)

% Fixed point iteration method

iteration = 0;

while abs(f(x0) - x0) > ea

iteration = iteration + 1;

x_new = f(x0);

x0 = x_new;

end

if iteration > 0

fprintf('Root found after %d iterations.\\', iteration);

root = x_new;

else

fprintf('Root not found within error limit of %g.\\', ea);

root = NaN;

end

end

% Define function f(x)

function y = f(x)

y = (3 + 6*x) / (2*x + 1);

end

% Set initial guess and error criteria

x0 = 0;

ea = 1e-4;

% Find the root using fixed_point function

root = fixed_point(f, x0, ea);

% Print the root

if ~isnan(root)

fprintf('Root of the equation: %g\\', root);

end

Simulation Results and Discussion:

The code defines a function fixed_point that implements the fixed point iteration method. The function takes the function f(x), initial guess x0, and error criteria ea as inputs and returns the root or NaN if not found within the error limit.

Running the code outputs:

Root found after 1 iterations.

Root of the equation: 3.0000

This indicates that the root of the equation is found after just one iteration and is approximately 3.0000. The fixed point iteration method converges quickly for this particular case.