Fast Element-Wise power to non-integer value

Question

Alfonso Silva 2023-2-23

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https://ww2.mathworks.cn/matlabcentral/answers/1917955-fast-element-wise-power-to-non-integer-value

评论： Walter Roberson 2023-3-2

I am working on a non-linear optimization problem using fmincon. My current setup delivers the best results so far in simulations.

Now I am working on optimizing my code and I have a bottle-neck in computation time in the expression:

(1-(1-x.^a).^b);

where 'x' is a vector of Nx1 observations, with N "medium-large" (> 30000 rows). Moreover, 'a' and 'b' are non-integer parameters > 0 (to be fitted)

f is the function that I am minimizing, thus, it's called by the optimizer several times (90000 in my current setting) resulting in a significant bottleneck in the overall procedure.

Are there any ways to improve the speed of this calculation? According to the code, profiler, this line is responsible for 25% of the total computation time in seconds. A replicable example is shown below:

%Example
N = 30000;
x = linspace(0,1,N)'; %vector of N x 1
a = 1.2;        %positive non-integer power
b = 0.3;        %positive non-integer power
f3 = @() (1-(1-x.^a).^b);
timeit(f3)
ans = 0.0029

Although it seems fast, if the function is called by the optimization procedure 90000 times, then the total time is around 0.0029*90000 = 261 seconds

Question:

Is there any way to make this computation faster? So far, I've tried

bsxfun, however it's included in the .^ command for newer releases
Increase maxNumCompThreads to 8
Relax optimization Tolerance levels
Tried other optimizers (such as lsqnonlin) but deliver worst estimates
write the expression as logarithm and then take exponential (it seems slower in preliminar trials)

Thank you for your comments

All best!

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Alfonso Silva 2023-2-23

@John D'Errico @Bruno Luong

Thank you for your replies and for your time. Indeed big problems require some time. I've managed to reduce the time by avoiding unncessary calculations when possible, but ultimately it's about waiting.

Thank you again!

Walter Roberson 2023-2-23

[I was wondering whether it would be effective to taylor f3 to a reasonable number of terms. I experimented with taylor around b = 0.5 (midpoint of the range), tested at 0.1 0.2 0.3 0.4... and the results for order 10 or order 20 were not encouraging.

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