Trying to make an Adams-Bashforth method with Richardson error estimate

%This program solves the initial value problem
%   y' = f(x,y), x0 <= x<= b, y(x0)=y0

%Initializing vaiables 
 %f'(x,y)= 
 f = @(x,y) cos(y).^2;              %derivative in question
 g = @(x) atan(x);                  %this is the actual solution
 x0 = 0;                             %initial value of x
 x_end = 10;                         %end of approximation
 h = 0.1;                           %size of decimal place (0.1,0.001,etc
 y0=0;                               %initial value of y
 
  n = fix((x_end-x0)/h)+1;
 x = linspace(x0,x_end,n);
 y = zeros(n,1);
 y(1) = y0;
 f1 = f(x(1),y(1));
 y(2) = y(1)+h*f1;
 
 %need to add error
 for i = 3:n 
     f2 = f(x(i-1),y(i-1));
     y(i) = y(i-1)+h*(3*f2-f1)/2;
     f1 = f2;
    
 fprintf('%5.4f  %11.8f\n', x(i), y(i));
 plot(x(i),y(i),'b.'); grid on; 
 fplot(g,[x0,x_end]);
 xlabel('x values'); ylabel('y values');
 hold on;
 end
 
 

I'm not sure how I would add the Richardson error to this code. I see the formula in my textbook, but don't understand how I would make it work. . Like I don't really know what that means. I understand the AB method for solving DefEqs, but not ther errors

[Torsten]

I'm confident that after reading this article

https://en.wikipedia.org/wiki/Richardson_extrapolation

you will know how Richardson extrapolation works.

Answer 1

[Mayank Sengar]

Go through https://en.wikipedia.org/wiki/Richardson_extrapolation

for understand Richardson expoitation, here is the pseudocode for it:

https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/12Differentiation/richardson/matlab.html#:~:text=Matlab%20may%20be%20used%20as,f(x)%20is%20defined.

There is also a generalized Richardson extrapolation routine in file exchange: https://www.mathworks.com/matlabcentral/fileexchange/24388-richardson-extrapolation