Effect of dx on errors of different approximations

Fourth_order_approx program code

function out = fourth_order_approx(x,dx);
analytical_derivative=((x^3*cos(x))-(sin(x)*3*x^2))/x^6;
fourth_order_equation=((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)
out=abs(fourth_order_equation-analytical_derivative);
end​

Second_order_approx program code

function out = second_order_approx(x,dx);
analytical_derivative=((x^3*cos(x))-(sin(x)*3*x^2))/x^6;
second_order_equation=(((sin(x+dx))/((x+dx)^3))-((sin(x-dx))/(x-dx)^3))/(2*dx);
out=abs(second_order_equation-analytical_derivative);
end​

First_order_approx program code

function out = first_order_approx(x,dx);
analytical_derivative=((x^3*cos(x))-(sin(x)*3*x^2))/x^6;
first_order_equation=(((sin(x+dx))/((x+dx)^3))-((sin(x)/x^3)))/(dx)
out=abs(first_order_equation-analytical_derivative);
end​

Sample code for effect of dx or error

close all
clear all
clc
dx=linspace(1/40,1/400,30);
x=pi/3;
for j = 1:length(dx)
  first_order_error(j) =first_order_approx(x,dx(j));
  second_order_error(j)=second_order_approx(x,dx(j));
  fourth_order_error(j)=fourth_order_approx(x,dx(j));
  end
plot(dx,fourth_order_error);
xlabel(\'dx\');
ylabel(\'fourth order error values\');
title(\'dx Vs fourth order error\');  

​

Sample output

Fourth order error

Second order error

First order error

Comparison between graphs:

For first order error, the graph increases linearly with increase in size of dx. For second order error the magnitude of slope of reduction of error with decerease in size of dx is higher compared to that of first order error. This magnitude is even greater for fourth order error when compared to second order error. Thus fourth order approximations are more accurate when compared to third order approximation which is better when compared to first order approximations