Taylor Table Method and Matlab Code

clear all

close all

clc

% Define Inputs

x = pi/3;

dx = linspace(pi/40,pi/4000,50);

% Error calculation of fourth order approximation of second derivative

% Using CDE,SRSDE,SLSDE calculate the error

for i = 1:length(dx)

analytical_derivative = -(2*exp(x)*sin(x));

central_difference_error(i) = central_approximation(x,dx(i));

skewed_right_side_difference_error(i)= skewed_right_side_approximation(x,dx(i));

skewed_left_side_difference_error(i)= skewed_left_side_approximation(x,dx(i));

end

% Ploting the graph to compare the error

loglog(dx,central_difference_error,\'color\',\'r\')

hold on

loglog(dx,skewed_right_side_difference_error,\'color\',\'b\')

hold on

loglog(dx,skewed_left_side_difference_error,\'color\',\'g\')

legend(\'central difference error\',\'skewed right side difference\',\'skewed left side difference\')

xlabel(\'dx\')

ylabel(\'error\')

function solution = central_approximation(x,dx);

% Analytical function

%f(x) = exp(x)*cos(x)

% f\'\'(x) = -(2*exp(x)*sin(x));

analytical_derivative = -(2*exp(x)*sin(x));

% central_difference =(-0.0833*f(x-2*dx)+1.3333*f(x-dx)-2.5*f(x)+1.3333*f(x+dx)-0.0833*f(x+2*dx))/(dx)^2

% Fourth order approximation of second order derivative using central difference

central_difference =(-0.0833*exp(x-2*dx)*cos(x-2*dx)+1.3333*exp(x-dx)*cos(x-dx)-2.5*exp(x)*cos(x)+1.3333*exp(x+dx)*cos(x+dx)-0.0833*exp(x+2*dx)*cos(x+2*dx))/(dx)^2;

solution = abs(analytical_derivative - central_difference);

end

function solution = skewed_left_side_approximation(x,dx);

% Analytical function

%f(x) = exp(x)*cos(x)

% seccond order derivative

% f\'\'(x) = -(2*exp(x)*sin(x))

analytical_derivative = -(2*exp(x)*sin(x));

% central_difference =(-0.8333*f(x-5*dx)+5.0833*f(x-4*dx)-13*f(x-3*dx)+17.8333*f(x-2*dx)-12.833*f(x-dx)+3.75*(x))/(dx)^2

% Fourth order approximation of second order derivative using central difference

skewed_left_side_difference =(-0.8333*exp(x-5*dx)*cos(x-5*dx)+5.0833*exp(x-4*dx)*cos(x-4*dx)-13*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)+3.75*exp(x)*cos(x))/(dx)^2;

solution = abs(analytical_derivative - skewed_left_side_difference);

end

function solution = skewed_right_side_approximation(x,dx);

% Analytical function

%f(x) = exp(x)*cos(x)

% seccond order derivative

% f\'\'(x) = -(2*exp(x)*sin(x))

analytical_derivative = -(2*exp(x)*sin(x));

% central_difference =(3.75*f(x)-12.3333*f(x+dx)+17.8333*f(x+2*dx)-13*f(x+3*dx)+5.0833*f(x+4*dx))-0.8333*f(x+5*dx))/(dx)^2

% Fourth order approximation of second order derivative using central difference

skewed_right_side_difference =(3.75*exp(x)*cos(x)-12.8333*exp(x+dx)*cos(x+dx)+17.8333*exp(x+2*dx)*cos(x+2*dx)-13*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;

solution = abs(analytical_derivative - skewed_right_side_difference);

end

we plot the graph taking the central approximation(five variable), skewed right side approximation (six variable), a skewed left side approximation (six variable). As we can see in the graph that the error in skewed right side differences maximum when the value of dx is greater but when the value of dx decreases, the error also decreases and come to a constant value.

The error in skewed left side differences maximum when the value of dx is greater but when the value of dx decreases and after some value of dx it again increases and comes to a constant value.The error in the central difference is approximately constant at every value of dx we choose.

we can see that in CDS we require the values of forwarding and backward points and compute it at the same time so the effective points are not considered which are located in corners. whereas in a skewed scheme we can easily calculate the variable at the corners (either for the right or left scheme)