Good afternoon, Ralf.

We have done some of the measurements you recommended, for your convenience we 
have created a separate folder with notebooks where we measured memory usage 
and performance of our interpretation against Scipy. Separately you can run the 
tests on your hardware and separately measure memory. I've left the link below.

https://github.com/2D-FFT-Project/2d-fft/tree/main/notebooks

We measured efficiency for 4 versions - with multithreading and data type 
conversion. According to the results of the tests, our algorithm has the 
greatest lead in the case with multithreading and without data type conversion 
- 75%, the worst performance without multithreading and with data type 
conversion - 14%. In terms of memory usage we beat NumPy and Scipy by 2 times 
in all cases, I think this is a solid achievement at this point. 

I can generalise that our mathematical approach still has a serious advantage, 
nevertheless we lose always to Scipy in inverse operation case, we haven't 
figured out the reasons yet, we are discussing it at the moment, but we will 
fix it. 

It is important to note that at this stage our algorithm shows the above 
perfomance on matrices of size powers of two. 
This is a specificity of the mathematical butterfly formula. We are 
investigating ways to remove this limitation, we already assessed the effect of 
element imputation and column dropping, the result is not accurate enough. 
Otherwise, we can suggest putting our version to work only in cases of the 
mentioned matrices, it'll still be an upgrade for NumPy.

At this point I can say that we are willing to work and improve the existing 
version within our skills, knowledge and available resources. We still live 
with the idea of adding our interpretation or idea to the existing NumPy 
package, as in theoretical perspective within the memory usage and efficiency, 
it can give a serious advantage on other projects built on NumPy. 

Thank you for your time, we will continue our work and look forward to your 
review.
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