Discretization basics

Objective :-

Function programe :-

1.First order approximation :-

% function for first order error calculation
function output1 = first_order_approximate(x,dx)
 analytical_derivative = (x^3*(cos(x))- sin(x)*3*x^2)/x^6;
 num_first_order = (sin(x+dx)/(x+dx)^3-sin(x)/x^3)/dx;
 output1 = abs(analytical_derivative -  num_first_order);
end

• In the 1st order approximation method forward differencing will taken from one forward point to calculate the error value of the function.

2.Second order approximation :-

% function for second order error calculation
function output2 = second_order_approximate(x,dx)
 analytical_derivative = (x^3*(cos(x))- sin(x)*3*x^2)/x^6;
 num_second_order = (sin(x+dx)/(x+dx)^3-sin(x-dx)/(x-dx)^3)/(2*dx);
 output2 = abs(analytical_derivative - num_second_order);
end

• In the second order approximation method center differencing wil taken from analytical derivation point to calculate the error value of the function.

3.Fourth order approximation :-

% function for fourth order error calculation
function output3 = fourth_order_approximate(x,dx)
 analytical_derivative = (x^3*(cos(x))- sin(x)*3*x^2)/x^6;
 num_fourth_order = (sin(x-2*dx)/(x-2*dx)^3-8*(sin(x-dx)/(x-dx)^3)+8*(sin(x+dx)/(x+dx)^3)-(sin(x+2*dx)/(x+2*dx)^3))/(12*dx);   
 output3 = abs(analytical_derivative - num_fourth_order);
end

• In the fourth order approximation method we have to calculate the error from analytical derivative to fourth order approximation then error value will be finded over.

Main programe :-

% Programe for discretization basics
clear all
close all
clc

% analytical function fx = sin(x)/x^3
% analytical derivative f'(x) = (x^3*(cos(x))- sin(x)*3*x^2)/x^6;
x = pi/3;
dx = linspace(pi/4,pi/4000,20);

for i = 1:length(dx)
    error1(i) = first_order_approximate(x,dx(i));
    error2(i) = second_order_approximate(x,dx(i));
    error3(i) = fourth_order_approximate(x,dx(i));
end

errorgraph = [error1(15) error2(15) error3(15)];
a = categorical({'1st_o_approx','2nd_o_approx','4th_o_approx'});
bar(a,errorgraph);
title('Comparison of first,second and fourth order approximation');
ylabel('error');
xlabel('approximation');

Explanation :-

1. In main programe we have to call function for first, second , fourth order approximation
2. After that we have to create the analytical function of fx = sin(x)/x^3.
3. Then create analytical derivative f'(x) = (x^3*(cos(x))- sin(x)*3*x^2)/x^6.
4. we have to compute the first derivative at x = pi/3 , ather that we have to create linspace command .
5. Create for loop command to analysis the error .
6. By using Bar command we have to create the bar value error.

Error while facing in programe :-

No error

Result :-

Bar Graph :