Skip to main content

Python functions for using Chebyshev polynomials and (real, 2D, orthonormal) spherical harmonics

Project description

orthopoly

This is a package for using sets of orthogonal functions/polynomials. Currently, it includes the Chebyshev polynomials and the spherical harmonics. The package is MIT licensed. Use it however you like. Documentation is available here: https://wordsworthgroup.github.io/orthopoly

Installing/Using

To install the package, you can

> pip install orthopoly

or you can download/clone the repository, put the top directory in your sys.path, then import it.

chebyshev

The chebyshev module is pretty well developed. It includes many functions for evaluating the Chebyshev polynomials and their derivatives. The module also includes higher level functions for setting up the elements needed to solve boundary value problems (using the pseudospectral method), interpolate using a Chebyshev grid, and perform the spatial discretization of a PDE solver. Generally, these methods are very well suited to smooth problems.

For performing a discrete Chebyshev transform (generating a Chebyshev expansion from a set of points in 1D), the cheby_coef_setup function can be used. It allows one of the boundary conditions to be the value of the expansion's first derivative and returns a matrix allowing expansion coefficents to be computed by solving a linear system. When there are no derivatives, the transform can be computed (on the appropriate cheby_grid) with a discrete cosine transform (DCT). The cheby_dct function does this and cheby_dct_setup sets up the grid as well.

Information about the Chebyshev polynomials is widely available, but a few particularly helpful references are below. The Boyd book is especially good.

  • Boyd, John P. Chebyshev and Fourier spectral methods. Courier Corporation, 2001.
  • Fornberg, Bengt. A practical guide to pseudospectral methods. Vol. 1. Cambridge university press, 1998.
  • Canuto, Claudio, et al. Spectral methods. Springer-Verlag, Berlin, 2006.

spherical_harmonic

The spherical_harmonic module provides functions for evaluating the real, two-dimensional (surface), orthonormal, spherical harmonics. It contains functions for evaluating the associated Legendre polynomials and their first two derivatives via stable recursion relationships. From the associated Legendre polynomials, the spherical harmonics, their gradients, and their Laplacians can be evaluated. The module also contains some functions for creating grids on the sphere (regular, icosahedral, and Fibonacci) and for creating random spherical harmonic expansions with specific power density relationships (noise). The module does not have functions for performing spherical harmonic analysis (transforming from values on the sphere to expansion coefficients).

For some applications, fitting a spherical harmonic expansion to data in spherical coordinates is useful. A least squares fit can be computed with the pseudoinverse of a matrix full of spherical harmonic function evaluations (see sph_har_matrix and related functions). However, this should only be done when the number of points is much greater than the number of terms in the fitted expansion.

The books cited above have some good discussion of spherical harmonics. Other useful sources include:

  • Press, William H., et al. Numerical recipes 3rd edition: The art of scientific computing. Cambridge university press, 2007.
  • Dahlen, F., and Jeroen Tromp. Theoretical global seismology. Princeton university press, 1998.
  • Bosch, W. "On the computation of derivatives of Legendre functions." Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy 25.9-11 (2000): 655-659.

Project details


Download files

Download the file for your platform. If you're not sure which to choose, learn more about installing packages.

Source Distribution

orthopoly-0.4.tar.gz (16.9 kB view details)

Uploaded Source

Built Distribution

orthopoly-0.4-py3-none-any.whl (20.8 kB view details)

Uploaded Python 3

File details

Details for the file orthopoly-0.4.tar.gz.

File metadata

  • Download URL: orthopoly-0.4.tar.gz
  • Upload date:
  • Size: 16.9 kB
  • Tags: Source
  • Uploaded using Trusted Publishing? No
  • Uploaded via: twine/1.13.0 pkginfo/1.5.0.1 requests/2.22.0 setuptools/41.2.0 requests-toolbelt/0.9.1 tqdm/4.32.1 CPython/3.6.9

File hashes

Hashes for orthopoly-0.4.tar.gz
Algorithm Hash digest
SHA256 ebcc143a514a5aac3a15f06a21c553e7050201ad1facdf8ba0486c50af3d7455
MD5 b2f12bff17554032bdd4ae9346213ace
BLAKE2b-256 db5368c700d4ff3087ebb195dd855d03fc8c9a6df0fc305de885adab3eb69611

See more details on using hashes here.

File details

Details for the file orthopoly-0.4-py3-none-any.whl.

File metadata

  • Download URL: orthopoly-0.4-py3-none-any.whl
  • Upload date:
  • Size: 20.8 kB
  • Tags: Python 3
  • Uploaded using Trusted Publishing? No
  • Uploaded via: twine/1.13.0 pkginfo/1.5.0.1 requests/2.22.0 setuptools/41.2.0 requests-toolbelt/0.9.1 tqdm/4.32.1 CPython/3.6.9

File hashes

Hashes for orthopoly-0.4-py3-none-any.whl
Algorithm Hash digest
SHA256 2ef854653dbe703b53fb055c4c770eda83972bef814907711ce9509fcd641701
MD5 a156562d8eefbb567525eb3f44efada0
BLAKE2b-256 66f6fa95f5899fb92e8a4763c9c0e1fc37d129da67e98a5de62a542ad571aba5

See more details on using hashes here.

Supported by

AWS Cloud computing and Security Sponsor Datadog Monitoring Fastly CDN Google Download Analytics Pingdom Monitoring Sentry Error logging StatusPage Status page