other

.

Bisection method

clear

clc

f = @(x) cos(x) - x.*exp(x);

a = 0;

b = 1;

n = 1e4;

error = 10.^(-4);

c_values = zeros(n,1);

i = 0;

while (b-a)/2 > error

i = i + 1;

c = (a+b)/2;

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

b = c;

else

a = c;

end

i = i + 1;

c_values(i) = c;

end

c_values = c_values(1:i);

root = (a+b)/2;

fprintf('The approximate root is %d \n', root);

fprintf('Number of iterations %d\n', i)

Newton–Raphson Method

% Newton-Raphson method

clc

clear

g = @(x) cos(x) - x*exp(x); % function

dg = @(x) -sin(x) - exp(x) - x*exp(x); % derivative of the function

x0 = 1; % initial point

error_nr = 10^(-4);

while true

x1 = x0 - g(x0)/dg(x0);

if abs(x1 - x0) < error_nr

break;

end

x0 = x1;

end

fprintf('the value of x0 %d\n', x0);

Secant Method

clc

clear

f = @(x) cos(x) - x.*exp(x);

x_0 = 0;

x_1 = 1;

error_2 = 10^(-4);

while abs(x_1 - x_0) > error_2

x_2 = x_1 - f(x_1)*(x_1 - x_0)/(f(x_1) - f(x_0));

x_0 = x_1;

x_1 = x_2;

end

root_secant = x_1

Fixed Point Iteration

clear

clc

f = @(x) cos(x) - x.*exp(x);

g = @(x) cos(x)./exp(x);

x0 = 0.5;

error_4 = 10^(-4);

while true

x1 = g(x0);

if abs(x1 - x0) < error_4

break;

end

x0 = x1;

end

root_fixed_point = x1

Regula Falsi Method

h = @(x) cos(x) - x.*exp(x);

c = 0;

d = 1;

error_3 = 10^(-4);

while true

x = (c*h(d) - d*h(c)) / (h(d) - h(c));

if abs(h(x)) < error_3

break;

end

if h(c)*h(x) < 0

d = x;

else

c = x;

end

end

root_regula_falsi = x;

2nd question

f1 = @(x,y) y*cos(x*y) + 1;

f2 = @(x,y) sin(x*y) + x - y;

df1dx = @(x,y) -y^2*sin(x*y);

df1dy = @(x,y) cos(x*y) - x*y*sin(x*y);

df2dx = @(x,y) y*cos(x*y) + 1;

df2dy = @(x,y) x*cos(x*y) - 1;

x = 1;

y = 2;

error_5 = 10^(-4);

while true

F = [f1(x,y); f2(x,y)];

J = [df1dx(x,y), df1dy(x,y);

df2dx(x,y), df2dy(x,y)];

J_inv = inv(J);

delta = J_inv * F;

x_new = x - delta(1);

y_new = y - delta(2);

if max(abs([x_new - x, y_new - y])) < error_5

break;

end

x = x_new;

y = y_new;

end

x_solution = x_new;

y_solution = y_new;

s = f1(x_new, y_new);