COMPUTING HIGHER ORDER APPROXIMATIONS

OBJECTIVE:

To Derive the following 4th order approximations of the second order derivative

1. Central difference

2. Skewed right sided difference

3. Skewed left sided difference

Once this is done, we write a program in Matlab to evaluate the second order derivative of the analytical function exp(x)*cos(x) and compare it with the 3 numerical approximations that we have derived.

a. To Provide a plot that compares the absolute error between the above mentioned schemes

b. To explain how these errors compare with FDS and BDS approximations

THE CODE:

% Taylors Table Derivative

% Analytical Way

clc
clear all
close all
x=pi/3;
analytical=-2*exp(x)*sin(x);
n=10;
dx=linspace(0.01,1,10);
% Using Forward Difference Scheme
for i=1:length(dx ) 
forward(i)=((35*f(x))-(104*f(x+dx(i)))+(114*f(x+(2*dx(i))))-(56*f(x+(3*dx((i)))))+(11*f(x+(4*dx(i)))))/(12*(dx(i))^2);
backward(i)=((35*f(x))-(104*f(x-dx(i)))+(114*f(x-(2*dx(i))))-(56*f(x-(3*dx((i)))))+(11*f(x-(4*dx(i)))))/(12*(dx(i))^2);
cds(i)=((-30*f(x))+(16*f(x-dx(i)))+(16*f(x+(dx(i))))-(f(x+(2*dx((i)))))-(f(x-(2*dx(i)))))/(12*(dx(i))^2); 
error_forward(i)=abs(forward(i)-analytical);
error_backward(i)=abs(backward(i)-analytical);
error_cds(i)=abs(cds(i)-analytical);
end
loglog(dx,error_forward,'-kd','markerfacecolor','r','markersize',7)
grid on
hold on
loglog(dx,error_backward,'-ko','markerfacecolor','b','markersize',7)
hold on
loglog(dx,error_cds,'-ks','markerfacecolor','g','markersize',7)
fprintf(' \n Forward scheme= %g',forward(n));
fprintf(' \n backward scheme = %g',backward(n));
fprintf(' \n cds scheme = %g',cds(n));
fprintf(' \n Actual derivative = %g',analytical);
fprintf(' \n Forward Error = %g',error_forward(n));
fprintf(' \n Backward Error = %g',error_backward(n));
fprintf(' \n CDS Error= %g',error_cds(n));
xlabel('step size')
ylabel('error')
legend('SKEWED RIGHT','SKEWED LEFT','CDS','location','northwest')
title('step size vs relative error');

fname={'forward','backward','cds'};
figure(2)
x=[1:3];
y=[error_forward(n) error_backward(n) error_cds(n)];
bar(x,y)
set(gca,'xTick',1:length(fname),'xTickLabel',fname);
xlabel('order')
ylabel('error')
legend('error prediction using 1 2 4 order approximation')

FUNCTION CODE:

function y=f(x);

y=exp(x)*cos(x);

end

THE RESULTS:

Forward scheme= 47.5215 
backward scheme = -2.59705 
cds scheme = -5.17395 
Actual derivative = -4.93575 
Forward Error = 52.4573 
Backward Error = 2.3387 
CDS Error= 0.238207

The above program calculates analytical value of the function at a user defined point (user needs to provide the input for the point where apporximation is to be calculated, max step size, min step size and the interval size between the point of consideration).

The value of x and dx is provided to three functions,i.e., central approximation, right skewed  approximation and left skewed approximation. The values from these variables are considered and plotted on a graph.

The graph obtained is shown above. From the graph we can clearly understand that Central Approximation Error is less than Left Approximation Error followed by Right Approximation Error. Hence the vartiation of the error is as follows:

Central Approximation Error < Left Approximation Error < Right Approximation Error.

Thus it is prefered to used central differencing scheme to solve any given equation to obtain more accurate answers.