Week 3.5 - Deriving 4th order approximation of a 2nd order derivative using Taylor Table method

clc
x = pi/3;

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

clock = 0;
%Defining the for loop for storing the function value for each x value
for n = 40:40:4000

  clock = clock+1;
  i = clock;
  dx=pi/n;
  dx_value(i) = dx;
  %Analtical Calculation
  ##  f_dash = (exp(x)*cos(x))+(-sin(x)*exp(x));
  f_double_dash(i) = (((exp(x)*cos(x))+(-sin(x)*exp(x)))+((-sin(x)*exp(x))+(-cos(x)*exp(x))));
  f_double_dash(i)
  %central_difference
  %Using Taylor's table Method, we found the coordinates

  a=-0.083333;
  b=1.333333;
  c=-2.500000;
  d=1.333333;
  e=-0.083333;

  central_differencing_fourth_order_approx(i) = ((a*exp(x-(2*dx))*cos(x-2*dx))+(b*exp(x-dx)*cos(x-dx))+(c*exp(x)*cos(x))+(d*exp(x+dx)*cos(x+dx))+(e*exp(x+2*dx)*cos(x+2*dx)))/dx^2;
  error_central_differencing(i) = abs(central_differencing_fourth_order_approx(i)-f_double_dash(i));

  %skewed left-sided difference
  %Using Taylor's table Method, we found the coordinates
  a=-0.83333;
  b=5.08333;
  c=-13.00000;
  d=17.83333;
  e=-12.83333;
  g=3.75000;

  skewed_left_sided_fourth_order_approx(i)=((a*exp(x-5*dx)*cos(x-5*dx))+(b*exp(x-4*dx)*cos(x-4*dx))+(c*exp(x-3*dx)*cos(x-3*dx))+(d*exp(x-2*dx)*cos(x-2*dx))+(e*exp(x-dx)*cos(x-dx))+(g*exp(x)*cos(x)))/dx^2;
  error_skewed_left_sided(i)=abs(skewed_left_sided_fourth_order_approx(i)-f_double_dash(i));


  %skewed right-sided
  %Using Taylor's table Method, we found the coordinates
  a=3.75000;
  b=-12.83333;
  c=17.83333;
  d=-13.00000;
  e=5.08333;
  g=-0.83333;

  skewed_right_sided_fourth_order_approx(i)=((a*exp(x)*cos(x))+(b*exp(x+dx)*cos(x+dx))+(c*exp(x+2*dx)*cos(x+2*dx))+(d*exp(x+3*dx)*cos(x+3*dx))+(e*exp(x+4*dx)*cos(x+4*dx))+(g*exp(x+5*dx)*cos(x+5*dx)))/dx^2;
  error_skewed_right_sided(i)=abs(skewed_right_sided_fourth_order_approx(i)-f_double_dash(i));

endfor

%Ploting the value of the analtical, first, second and fourth order to understand the deveation
semilogx(dx_value,f_double_dash);
hold on;
semilogx(dx_value,central_differencing_fourth_order_approx,'r');
hold on;
semilogx(dx_value,skewed_left_sided_fourth_order_approx,'b');
hold on;
semilogx(dx_value,skewed_right_sided_fourth_order_approx,'g');
xlabel('dx');
ylabel('error value');
title('4th order error of second-order derivative using central, skewed right-sided, and skewed left-sided scheme');
legend('f dash analtical', 'Central Difference Method','Skewed Right-sided Error', 'Skwed Left-sided Error');
print -djpg '4th order error of second-order derivative using central, skewed right-sided, and skewed left-sided scheme.jpg'