Why am I obtaining incorrect values for Lipschitz 1/2 norms using the central difference method?
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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?
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Réponses (2)
Bjorn Gustavsson
le 22 Déc 2022
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)
HTH
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Bora Eryilmaz
le 22 Déc 2022
Unless I misunderstodd something, the derivative of
is
.
Within the domain of [0,1], the maximum absolute value of this is +Inf, not 2.
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