% Problem 1:
clear all; clc; close all;

% Data
x = [0 10 15 20 22.5 30];  % x values
fx = [0 227.04 362.78 517.35 602.97 901.67];  % corresponding y values
x_new = 16;  % x value for interpolation

% Calculate first-order polynomial interpolation using Newton's divided difference
fx0 = fx(3); fx1 = fx(4);
x0 = x(3); x1 = x(4);

% Divided differences
f0 = fx0;
f1 = (fx1 - fx0) / (x1 - x0);

% Evaluate the interpolated y value using Newton's interpolation formula
fx_new = f0 + (x_new - x0) * f1;

% Plot original data and interpolated point using Newton Interpolation
figure
plot(x, fx, 'bo-', x_new, fx_new, 'rx')
grid on, xlabel('x data'), ylabel('y data'), title('First-Order Polynomial Interpolation')
legend('Data', 'Interpolated Point', 'location', 'northwest')


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% Problem 2:
clear all; clc; close all;

% Data
x = [0 10 15 20 22.5 30];
fx = [0 227.04 362.78 517.35 602.97 901.67];
x_new = 16;  % x value for interpolation

% Calculate second-order polynomial interpolation using Newton's divided difference
fx0 = fx(2); fx1 = fx(3); fx2 = fx(4);
x0 = x(2); x1 = x(3); x2 = x(4);

% Divided differences
f0 = fx0;
f1 = (fx1 - fx0) / (x1 - x0);
f2 = ((fx2 - fx1) / (x2 - x1) - f1) / (x2 - x0);

% Evaluate the interpolated y value using Newton's interpolation formula
fx_new = f0 + (x_new - x0) * f1 + (x_new - x0) * (x_new - x1) * f2;

% Plot original data and interpolated point
figure
plot(x, fx, 'bo-', x_new, fx_new, 'rx')
grid on, xlabel('x data'), ylabel('y data'), title('Second-Order Polynomial Interpolation')
legend('Data', 'Interpolated Point', 'location', 'northwest')


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% Problem 3:

clear all; clc; close all;

% Data
x = [0 10 15 20 22.5 30];
fx = [0 227.04 362.78 517.35 602.97 901.67];
x_new = 16;  % x value for interpolation

% Calculate third-order polynomial interpolation using Newton's divided difference
fx0 = fx(2); fx1 = fx(3); fx2 = fx(4); fx3 = fx(5);
x0 = x(2); x1 = x(3); x2 = x(4); x3 = x(5);

% Divided differences
f0 = fx0;
f1 = (fx1 - fx0) / (x1 - x0);
f2 = ((fx2 - fx1) / (x2 - x1) - f1) / (x2 - x0);
f3 = (((fx3 - fx2) / (x3 - x2) - (fx2 - fx1) / (x2 - x1)) / (x3 - x1) - f2) / (x3 - x0);

% Evaluate the interpolated y value using Newton's interpolation formula
fx_new = f0 + (x_new - x0) * f1 + (x_new - x0) * (x_new - x1) * f2 + ...
    (x_new - x0) * (x_new - x1) * (x_new - x2) * f3;

% Plot original data and interpolated point
figure
plot(x, fx, 'ko-', x_new, fx_new, 'rx')
grid on, xlabel('x data'), ylabel('y data'), title('Third-Order Polynomial Interpolation')
legend('Data', 'Interpolated Point', 'location', 'northwest')