bisection

b

clear;

close all;

% Define function

f = @(x) 3*x - cos(x) - 1;

% Initial interval [a,b]

a = 0;

b = 1;

% Check validity

if f(a)*f(b) >= 0

error('f(a) and f(b) must have opposite signs');

end

% Tolerances

tol_x = 1e-6; % interval tolerance

tol_f = 1e-6; % function tolerance

maxIter = 50;

fprintf('Iter\t a\t\t b\t\t c\t\t |b-a|\t\t |f(c)|\n');

for n = 1:maxIter

% Midpoint

c = (a + b)/2;

fprintf('%d\t %.6f\t %.6f\t %.6f\t %.6e\t %.6e\n', ...

n, a, b, c, abs(b-a), abs(f(c)));

% Stopping criteria

if abs(b - a) < tol_x || abs(f(c)) < tol_f

break;

end

% Interval update

if f(a)*f(c) < 0

b = c;

else

a = c;

end

end

fprintf('\nApproximate root = %.8f\n', c);

fprintf('Iterations = %d\n', n);

fixed point

clc;

clear;

close all;

% Define functions

f = @(x) 3*x - cos(x) - 1;

g = @(x) (cos(x) + 1)/3;

% Initial guess

x0 = 0.5;

% Tolerances

tol_x = 1e-12; % iteration tolerance

tol_f = 1e-12; % function tolerance

maxIter = 50;

fprintf('Iter\t x_n\t\t |x_n-x_{n-1}|\t |f(x_n)|\n');

for n = 1:maxIter

x1 = g(x0);

fprintf('%d\t %.8f\t %.6e\t %.6e\n', ...

n, x1, abs(x1-x0), abs(f(x1)));

% Stopping criteria

if abs(x1 - x0) < tol_x || abs(f(x1)) < tol_f

break;

end

x0 = x1;

end

fprintf('\nApproximate root = %.8f\n', x1);

fprintf('Iterations = %d\n', n);

newton raphson

clc;

clear;

close all;

% Define function and derivative

f = @(x) 3*x - cos(x) - 1;

df = @(x) 3 + sin(x);

% Initial guess

x0 = 0.5;

% Tolerances

tol_x = 1e-6; % iteration tolerance

tol_f = 1e-6; % function tolerance

maxIter = 50;

fprintf('Iter\t x_n\t\t |x_n-x_{n-1}|\t |f(x_n)|\n');

for n = 1:maxIter

x1 = x0 - f(x0)/df(x0);

fprintf('%d\t %.8f\t %.6e\t %.6e\n', ...

n, x1, abs(x1-x0), abs(f(x1)));

% Stopping criteria

if abs(x1 - x0) < tol_x || abs(f(x1)) < tol_f

break;

end

x0 = x1;

end

fprintf('\nApproximate root = %.8f\n', x1);

fprintf('Iterations = %d\n', n);

regular falsi

clc;

clear;

close all;

% Function

f = @(x) 3*x - cos(x) - 1;

% Initial interval [a,b] with f(a)*f(b) < 0

a = 0;

b = 1 ;

% Parameters

iter = 10;

tol = 1e-12;

fprintf('iter\t\t a\t\t b\t\t c\t\t f(c)\n');

for i = 1:iter

% False position formula

c = (a*f(b) - b*f(a)) / (f(b) - f(a));

fprintf('%d\t %0.8f\t %0.8f\t %0.8f\t %0.15e\n', ...

i, a, b, c, f(c));

% Convergence check

if abs(f(c)) < tol

break;

end

% Interval update (bracketing preserved)

if f(a)*f(c) < 0

b = c;

else

a = c;

end

end

fprintf('\nApproximate root = %.12f\n', c);

secant method

clc;

clear;

close all;

% Define function

f = @(x) 3*x - cos(x) - 1;

% Initial guesses (twostarting points)

x0 = 0.0;

x1 = 1.0;

% Tolerances

tol_x = 1e-6; % iteration tolerance

tol_f = 1e-6; % function tolerance

maxIter = 50;

fprintf('Iter\t x_n\t\t |x_n-x_{n-1}|\t |f(x_n)|\n');

for n = 1:maxIter

% Secant formula

x2 = x1 - f(x1)*(x1 - x0)/(f(x1) - f(x0));

fprintf('%d\t %.8f\t %.6e\t %.6e\n', ...

n, x2, abs(x2 - x1), abs(f(x2)));

% Stopping criteria

if abs(x2 - x1) < tol_x || abs(f(x2)) < tol_f

break;

end

% Update points

x0 = x1;

x1 = x2;

end

fprintf('\nApproximate root = %.8f\n', x2);

fprintf('Iterations = %d\n', n);