Discretization Basics

Its OCTAVE CODE.


clear all;
close all;
clc;

%analytical function=sin(x)/x^3;

%analytical derative:-

%  f\'(x)= ((x^3*(cos(x)))-(sin(x)^3*x^2))/x^6;

x=pi/3;

analytical_derivative=((x^3*(cos(x)))-(sin(x)*3*x^2))/x^6;

%Numerical Derivative

%Forward differencing(First Order Approximation)

%(F(x+dx)-F(x))/dx

dx=pi/40000,

forward_differencing=(((sin(x+dx))/(x+dx)^3)-((sin(x))/x^3))/dx;

%central approximation(2nd order approximation)

Central_differencing=(((sin(x+dx))/(x+dx)^3)-(sin(x-dx))/((x-dx)^3))/(2*dx);

%Fourth Order Approximation

z=(sin(x-2*dx))/(x-2*dx)^3;
a=(sin(x-1*dx))/(x-1*dx)^3*(-8);
q=(sin(x+1*dx))/(x+1*dx)^3*8;
w=(sin(x+2*dx))/(x+2*dx)^3*(-1);

Fourth_order_approximation=(z+a+q+w)/(12*dx);


error1=abs(forward_differencing-analytical_derivative);
error2=abs(Central_differencing-analytical_derivative);
error3=abs(Fourth_order_approximation-analytical_derivative);

%Bar graph for difference between approximations and anaylitical derivative(error)
figure(1)
y=[forward_differencing analytical_derivative;Central_differencing analytical_derivative;Fourth_order_approximation analytical_derivative];
bar(y)
legend(\'Approximations\',\'analytical answer\');