Week 3.5 - Deriving 4th order approximation of a 2nd order derivative using Taylor Table method
main code
clear all
close all
clc
% Input value for x
x=pi/3;
dx = linspace(pi/4,pi/4000,25);
for i=1:length(dx)
central_diff_error(i) = fourth_order_centra_diff(x,dx(i));
skewed_right_side_error(i)= skewed_right_side(x,dx(i));
skewed_left_side_error(i)= skewed_left_side(x,dx(i));
end
figure(1)
loglog(dx,central_diff_error,'b')
hold on
loglog(dx,skewed_right_side_error,'g')
hold on
loglog(dx,skewed_left_side_error,'r')
xlabel('dx')
ylabel('error')
legend('central diff error','skewed right side error','skewed left side error')Central_differencing_error code
function Central_differencing_error = fourth_order_centra_diff(x,dx)
% Ana function = exp(x)*cos(x)
%ana dervative = -2*exp(x)*sin(x)
analatical_dervative = -2*exp(x)*sin(x);
% Central_differencing
central_diff_error = (((-0.0833)*((exp(x-(2*dx)))*(cos(x-(2*dx)))))+((1.3333)*((exp(x-dx))*(cos(x-dx))))-((2.5000)*((exp(x))*(cos(x))))+((1.3333)*((exp(x+dx))*(cos(x+dx))))-((0.0833)*((exp(x+(2*dx)))*(cos(x+(2*dx))))))/(dx^2);
% Central_differencing_error
Central_differencing_error = abs(central_diff_error-analatical_dervative);
endRight_sided_Skew_error code
function Right_sided_Skew_error = skewed_right_side(x,dx)
% Ana function = exp(x)*cos(x)
%ana dervative = -2*exp(x)*sin(x)
analatical_dervative = -2*exp(x)*sin(x);
% Fourth_order_Right_sided_differencing
skewed_right_side_error = (((3.7500)*((exp(x))*(cos(x))))-((12.8333)*((exp(x+dx))*(cos(x+dx))))+((17.8333)*((exp(x+(2*dx)))*(cos(x+(2*dx)))))-((13.0000)*((exp(x+(3*dx)))*(cos(x+(3*dx)))))+((5.0833)*((exp(x+(4*dx)))*(cos(x+(4*dx)))))-((0.8333)*((exp(x+(5*dx)))*(cos(x+(5*dx))))))/(dx^2);
% error
Right_sided_Skew_error = abs(skewed_right_side_error - analatical_dervative);
endLeft_sided_Skew_error code
function Left_sided_Skew_error = skewed_left_side(x,dx)
% Ana function = exp(x)*cos(x)
%ana dervative = -2*exp(x)*sin(x)
analatical_dervative = -2*exp(x)*sin(x);
% Fourth_order_Left_sided_differencing
skewed_left_side_error = (((-0.8333)*((exp(x-(5*dx)))*(cos(x-(5*dx)))))+((5.0833)*((exp(x-(4*dx)))*(cos(x-(4*dx)))))-((13.0000)*((exp(x-(3*dx)))*(cos(x-(3*dx)))))+((17.8333)*((exp(x-(2*dx)))*(cos(x-(2*dx)))))-((12.8333)*((exp(x-dx))*(cos(x-dx))))+((3.7500)*((exp(x))*(cos(x)))))/(dx^2);
% error
Left_sided_Skew_error = abs(skewed_left_side_error-analatical_dervative);
endOutput-:
Expanation-:
From above graph central diff method shows the less error as compared to right and left skew method.
While consider accuracy, central diff method is useful because it take data from left and right side that is positive and negative so it is resulted most accurate and less error method.
The Skewed method is only supported when we dont have sufficient data for central method just like at the bpundr we can not use the central method.