Question

Coding: Use MATLAB to figure out the following problem, if you do not know how to use MATLAB then please do not answer. Coding is required for the exercise.

For f(x) = arctan(x), find its zeros by implimenting Newtons method and the Secant method in Matlab. (Hint: Use Newtons method to calculate x1 for Secant method)

Comment all code please since I would like to learn how to do this correctly in MATLAB. Thank you.

Answer 1

MATLAB Script:

close all
clear
clc

syms x
tol = 1e-3; % Error tolerance

f = atan(x); % Given function f(x)
df = diff(f); % f'(x)

% Plot the function to get initial estimates of the roots
x_vals = -5:0.001:5; % Range [-5, 5] with step size of 0.001
f_vals = subs(f, x_vals); % Evaluate f(x) at x = x_vals
plot(x_vals, f_vals), grid on % Plot f(x) and turn on the gridlines
xlabel('x'), ylabel('f(x)') % Label the axes
title('f(x) = atan(x)') % Add a title

% Newton's method
x0 = -1; % Initial guess
result = newton(f, df, x0, tol);
fprintf('Root (using Newton''s method), x = %.3f\n', result)

% Secant method
x1 = double(x0 - subs(f, x0)/subs(df, x0)); % Apply Newton update rule to find x1
result = secant(f, x0, x1, tol);
fprintf('Root (using secant method), x = %.3f\n', result)

function x = newton(f, df, x0, tol)
x = x0;
while true
x_ = x; % Save previous iteration's result
x = double(x - subs(f, x)/subs(df, x)); % Newton Update Rule
if abs(x_ - x) < tol % Termination condition
break;
end
end
end

function x = secant(f, a, b, tol)
x = [a b];
while true
xl = x(end-1); xu = x(end);
y = ( xl*subs(f, xu) - xu*subs(f, xl) ) / (subs(f, xu) - subs(f, xl)); % Secant method's root update
x = [x y];
if abs(x(end-1) - x(end)) < tol % Termination condition
break
end
end
x = x(end);
end

Plot:

Output:

Root (using Newton's method), x = 0.000
Root (using secant method), x = 0.000