Why am I obtaining incorrect values for Lipschitz 1/2 norms using the central difference method?

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Dylan Holloway
Dylan Holloway 2022-12-22
编辑： Jan 2023-1-7
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?
  1 个评论
Jan
Jan 2022-12-22
编辑：Jan 2022-12-22
Ran in:
I cannot confirm your statement, that Matlab online calculates 2 as output:
f = @(x) 2*sqrt(1 - x);
x = linspace(0, 1, 1000);
df = (f(x + 1e-8) - f(x - 1e-8)) / 2e-8;
lipschitz_norm = max(abs(df))
lipschitz_norm = 1.4142e+04
df(end)
ans = -1.0000e+04 + 1.0000e+04i

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回答（2 个）

Bjorn Gustavsson
Bjorn Gustavsson 2022-12-22
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
Bora Eryilmaz
Bora Eryilmaz 2022-12-22
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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