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Gumerov Expansion Coefficients

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Documentation: https://gumerov-expansion-coefficients.readthedocs.io

Source Code: https://github.com/34j/gumerov-expansion-coefficients

Multiple translation and rotation coefficients for the 3D Helmholtz Equation

Installation

Install this via pip (or your favourite package manager):

pip install gumerov-expansion-coefficients[cli,cuda]

Usage

from gumerov_expansion_coefficients import translational_coefficients

translational_coefficients(
    k * r, theta, phi, same=True, n_end=10
)  # (R|R) coefficients from 0 to 9 th degree
translational_coefficients(
    k * r, theta, phi, same=False, n_end=10
)  # (S|R) coefficients from 0 to 9 th degree

  • The definition of spherical harmonics are same as in [1]. Note that there are 3 other common definitions, and this definition differs from scipy.special.sph_harm_y for negative m.

\[ Y_n^m (\theta, \phi) := (-1)^m \sqrt{\frac{(2n+1)(n-\left|m\right|)!}{4 \pi (n+\left|m\right|)!}} P_n^{\left|m\right|} (\cos \theta) e^{i m \phi} \]
\[ R_n^m (kr, \theta, \phi) := j_n(kr) Y_n^m (\theta, \phi) \]
\[ S_n^m (kr, \theta, \phi) := h_n^{(1)}(kr) Y_n^m (\theta, \phi) \]

  • The return array is 2D array with shape (n_end**2, n_end**2).

  • The first axis is to be summed over, resulting in the elemenary solutions at the second axis.

  • The coefficient coressponding to the quantum numbers (n, m) is mapped to n**2 + (m % (2 * n + 1))-th index, while in [2] it is mapped to n * (n + 1) + m-th index.

References

  • [1] Gumerov, N. A., & Duraiswami, R. (2004). Recursions for the Computation of Multipole Translation and Rotation Coefficients for the 3-D Helmholtz Equation. SIAM Journal on Scientific Computing, 25(4), 1344–1381. https://doi.org/10.1137/S1064827501399705

  • [2] Gumerov, N. A., & Ramani, D. (2002年). Computation of scattering from N spheres using multipole reexpansion. The Journal of the Acoustical Society of America, 112(6), 2688–2701. https://doi.org/10.1121/1.1517253

Benchmark

gec benchmark
gec plot

timing_results

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Credits

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