Why am I obtaining incorrect values for Lipschitz 1/2 norms using the central difference method?
조회 수: 4 (최근 30일)
The Lipschitz 1/2 norm is defined as the maximum value of the absolute value of the derivative of the function over all points in the domain of the function. I have this code that can approximate this value for a given function:
% Define the function f
f = @(x) x.^2;
% Define the domain of the function
x = linspace(-1, 1, 1000);
% Compute the derivative of the function using the central difference method
df = (f(x+1e-8) - f(x-1e-8)) / (2*1e-8);
% Compute the Lipschitz 1/2 norm of the function
lipschitz_norm = max(abs(df));
Here, our function f and linspace for x are just an example.
I am trying to compute the norm for f = @(x) 2*sqrt(1-x), with x = linspace(0, 1, 1000). Or really, f = @(x) c*sqrt(1-x), where c is a real number. Theoretically, it's obvious that the norm for any of these functions is |c|, for a given c. Online, using this code with the example
f = @(x) 2*sqrt(1-x), with x = linspace(0, 1, 1000) gives lipschitz_norm = 2, as it should, but when I run the exact same code on MATLAB on my own, I get 1.4142e+04. I've tried numerous different examples, and my answers have yet to line up. Is there something going on on my end?
Bjorn Gustavsson 2022년 12월 22일
You've forgot to take the derivative of the function on-line, and you seem to take some kind of derivative on your own computer. Since the derivative of sqrt(1-x) is 1/2./sqrt(1-x) the max of the absolute goes towards infinity as x aproaches 1 from below.
(Don't worrt, we've all been there)