spectral-derivatives

Python code to take derivatives by spectral methods using the Chebyshev, Fourier, Legendre, and Bernstein bases.

https://github.com/pavelkomarov/spectral-derivatives

Science Score: 57.0%

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Repository

Python code to take derivatives by spectral methods using the Chebyshev, Fourier, Legendre, and Bernstein bases.

Basic Info

Host: GitHub
Owner: pavelkomarov
License: bsd-3-clause
Language: Jupyter Notebook
Default Branch: main
Homepage:
Size: 52.2 MB

Statistics

Stars: 4
Watchers: 1
Forks: 1
Open Issues: 0
Releases: 12

Created over 1 year ago · Last pushed 11 months ago

Metadata Files

Readme Funding License Citation

This repository is home to Python code that can take spectral derivatives using the Chebyshev, Fourier, Legendre, and Bernstein bases, grounded in some pretty elegant, deep math. That is, given a vector representing samples of a smooth function, the code returns numerical derivatives, indicating slope, curvature, etc. at the sample points of an interpolation built from the basis functions.

When using the Fourier basis, spectral derivatives require periodic boundaries, but the polynomial bases (fastest to fit being the Chebyshev basis) allow arbitrary boundaries, extending the method to a much wider class of functions (albeit best with nonuniform sampling).

This package can be useful any time you want to take a derivative numerically, such as for doing PDE simulations. For taking derivatives of noisy data, spectral methods naturally enable global filtering by weighting basis function contributions, although polynomial bases suffer some weakness.

Note these methods are best for situations when you don't know the generating function of your data; if the generator is known, then autodifferentiation tools like JAX are your friend.

Installation and Usage

The package is a single module containing derivative functions. To install, execute: shell python3 -m pip install spectral-derivatives or from the source code shell python3 -m pip install . You should now be able to ```python

from specderiv import * import numpy as np

xn = np.cos(np.arange(21) * np.pi / 20) # cosine-spaced, includes last point yn = np.sin(xn) # can be periodic or aperiodic on domain [a, b] dyn = chebderiv(yn, x_n, 1)

thn = np.arange(20) * 2*np.pi / 20 # equispaced, excludes last point yn = np.sin(th_n) # must be periodic on domain [a, b) dyn = fourierderiv(yn, thn, 1) ``` For further usage examples, including in higher dimension, see the Jupyter notebooks: Chebyshev, Fourier, Legendre, and Bernstein.

Note that for fastest and most accurate results you should use equispaced samples on an open periodic interval with fourier_deriv and cosine-spaced points with cheb_deriv. legendre_deriv and bern_deriv are slower. All methods support arbitrary domains by internally performing an affine transformation to some canonical domain.

All methods support differentiating multidimensional data along an axis.

References

Trefethen, N., 2000, Spectral Methods in Matlab, https://epubs.siam.org/doi/epdf/10.1137/1.9780898719598.ch8
Johnson, S., 2011, Notes on FFT-based differentiation, https://math.mit.edu/~stevenj/fft-deriv.pdf
Kutz, J.N., 2023, Data-Driven Modeling & Scientific Computation, Ch. 11, https://faculty.washington.edu/kutz/kutzbookv2.pdf
Breuer, K. & Everson, R., 1990, On the errors incurred calculating derivatives using Chebyshev polynomials, https://doi.org/10.1016/0021-9991(92)90274-3

Owner

Name: Pavel Komarov
Login: pavelkomarov
Kind: user
Location: Seattle, Salt Lake City, Melbourne FL, Atlanta, Mountain View, Bradenton
Company: University of Washington, BioIntelliSense, Miim/SafeX/Banjo, Northrop Grumman, Georgia Institute of Technology, Microsoft

Website: pavelkomarov.com
Repositories: 26
Profile: https://github.com/pavelkomarov

I like people, math, reading, getting lost in a problem, code when it isn't painful, skiing, climbing, riding my bike, music, and progress.

Citation (CITATION.md)

# Citing Spectral-Derivatives

If you use this project in publications, please cite the following:

	@misc{spectral-derivatives,
		title={Spectral derivatives software}, 
		author={Pavel Komarov}, 
		url={http://www.github.com/pavelkomarov/spectral-derivatives}, 
		note={Version 0.1},
		year={2024}}

If you wish to cite the mathematical explanation, use:

	@article{spectral-derivatives-math
		title={Spectral Derivatives},
		author={Pavel Komarov},
		url={https://pavelkomarov.com/spectral-derivatives/math.pdf},
		year={2024}}

GitHub Events

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Last Year

Create event: 20
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Release event: 18
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Watch event: 5
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Issue comment event: 43
Push event: 516
Fork event: 1

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Past Year

Issues: 15
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Average time to close issues: about 1 month
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pavelkomarov (15)

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Packages

Total packages: 1
Total downloads:
- pypi 40 last-month

Total dependent packages: 0
Total dependent repositories: 0
Total versions: 13
Total maintainers: 1

pypi.org: spectral-derivatives

Functions to take spectral derivatives with the Chebyshev and Fourier bases

Documentation: https://pavelkomarov.com/spectral-derivatives
License: BSD License
Latest release: 0.7.4
published over 1 year ago

Versions: 13
Dependent Packages: 0
Dependent Repositories: 0
Downloads: 40 Last month

Rankings

Dependent packages count: 9.8%

Average: 32.4%

Dependent repos count: 55.0%

Maintainers (1)

pavelkomarov