forward, backward, central finite differences with step sizes.

I have to develop a code that can differentiate functions by using forward, backward, and central finite difference approaches, and I need to use varying step sizes to make the program run at higher accuracies. I also need to show an analytical solution showing the exact solution. So far I have this:

clc; clear; close all;
Fun = @(x) x.^2; %function
dFun = @(x) 2.*x; %derivative
x=linspace(-3,3,101); %from x = -3 to 3
F=Fun(x);
h=x(2)-x(1);

xCentral=x(2:end-1); %Central Difference Approach
dFCentral=(F(3:end)-F(1:end-2))/(2*h);

xForward=x(1:end-1); %Forward Difference Approach
dFForward=(F(2:end)-F(1:end-1))/h;

xBackward=x(2:end); %Backward Difference Approach
dFBackward=(F(2:end)-F(1:end-1))/h;
plot(x,dFun(x)); %Exact solution
hold on
plot(xCentral,dFCentral,'r')
plot(xForward,dFForward,'k');
plot(xBackward,dFBackward,'g');
legend('Analytic','Central','Forward','Backward') %Plot everything

However, I dont know how to implement a step size into the program, how would I do this?

[ahmed]

use matlab code to find the approximation of the solution sinh-1(x) by using backward

Answer 1

[Davide Masiello]

I took the liberty to give you an example with a different function, because the one in your question has a constant slope and different step sizes produce the same result, hence it's difficult to visualize. Also, I did it only for the central difference case.

Fun = @(x) x.^3;                    % Function
dFun = @(x) 3*x.^2;                 % First order derivative
fplot(dFun,[-3,3],'.'),hold on      % Plot of first order derivative
h = [1 0.5 0.1];                    % Three different step sizes

for i = 1:3                         % Loops over step sizes
x = -3:h(i):3;                      % Creates x array depending on step size
xd  = x(2:end-1);                   % Removes boundaries (can't be approximated with central difference) 
dFc = (Fun(x(3:end))-Fun(x(1:end-2)))/(2*h(i)); % Central difference scheme
plot(x(2:end-1),dFc)                % Plot the result
end
legend('Analytic','h = 1','h = 0.5','h = 0.1','Location','best')