forward, backward, central finite differences with step sizes.

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Steven Deng
Steven Deng 2022-10-20
评论： ahmed 2023-5-20
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?

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回答（1 个）

Davide Masiello
Davide Masiello 2022-10-21
编辑：Davide Masiello 2022-10-21
Ran in:
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')

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