Discretization Basics

AIM:

To write a program that compares the first,second and fourth order approximations of the first derivative against the analytical or exact derivative.

OBJECTIVE:

1.The function that to compute if f(x) = sin(x)/x^3

2.To compute the first derivative at x = pi/3 for dx = pi/30.

3.To compute the derivative both numerically and analytically.

4. Define an error parameter as the absolute difference in numerical and exact derivative at x = pi/3

5. Plot a bar graph that shows the error.

GIVEN:

function f(x) = sinx/x^3

first derivative at x=pi/3

Analytical first derivative = (x^3*cosx-3*sinx*x^3)/x^6

Assuming infinitesimal quantity, dx=pi/15

Forward and central differencing methods are used to find the numerical derivative.

First order approximation, forward differencing = [f(x+a)-f(x)]/ΔΔx

Second order approximation, central differencing =[f(x+a)-f(x-a)]/2ΔΔx

Fourth order approximation, central differencing

STEPWISE EXPLANATION:

clear all
close all
clc

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

1. Always start the programe with clear all close all and clc command,
2. The analytical derivative is given which is stored as f'(x)
3. The x value is also given as pi/3
4. Now the analytical derivative is caluculated and stored in it

%forward differencing(1st order approximation)
%forward difference approximation = (f(x+dx)-f(x))/dx
%numerical derivative
dx = pi/30

%forward differencing
forward_differencing = ((sin(x+dx)/(x+dx)^3)-(sin(x)/(x)^3))

%(2nd order approximation) 
central_differencing=((sin(x+dx)/(x+dx)^3)-(sin(x-dx)/(x-dx)^3))/(2*dx)

% (4th order approximation)
fourth_order_approximation=((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)

1. The forward differencing approximation is done using the provided formula
2. The derivative dx is given as pi/30 and is stored
3. Now the forward differencing is done using the provided formula
4. The same procedure is repeated for the central differnecing whereas the formula changes
5. And finally the fourth order approximation is done using the given formula.

%errors
error_first_order=abs(forward_differencing - analytical_derivative); 
error_second_order=abs(central_differencing - analytical_derivative);
error_fourth_order=abs(fourth_order_approximation - analytical_derivative);

1. The absolute value of errors are calucluated for first,second and fourth orders.

% bar graph showing error
error = [error_first_order, error_second_order, error_fourth_order];
bar(error,0.5)
xlabel('Order of approximation')
ylabel('Truncation Error')
legend('order of derivative approximation errors')

1. Plotting the bar graph for tranucation error and the order of approximation
2. labels and legends are given for the respective variables

FINAL CODE:

clear all
close all
clc

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

%forward differencing(1st order approximation)
%forward difference approximation = (f(x+dx)-f(x))/dx
%numerical derivative
dx = pi/30

%forward differencing
forward_differencing = ((sin(x+dx)/(x+dx)^3)-(sin(x)/(x)^3))

%(2nd order approximation) 
central_differencing=((sin(x+dx)/(x+dx)^3)-(sin(x-dx)/(x-dx)^3))/(2*dx)

% (4th order approximation)
fourth_order_approximation=((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)

%errors
error_first_order=abs(forward_differencing - analytical_derivative); 
error_second_order=abs(central_differencing - analytical_derivative);
error_fourth_order=abs(fourth_order_approximation - analytical_derivative);


% bar graph showing error
error = [error_first_order, error_second_order, error_fourth_order];
bar(error,0.5)
xlabel('Order of approximation')
ylabel('Truncation Error')
legend('order of derivative approximation errors')

OUTPUT:

On X axis 1 is represented as first order approximation,2 as second order approximation and fourth order approximation respectively.

RESULT:

As it is observed that on increasing the order of derivative the error is decreasing from 1st order derivative to Fourth order derivative.