ysimonov / Associated-Legendre-Polynomials

Derivatives of Normalized Associated Legendre Polynomials

The program generates

1. Normalized Associated Legendre Polynomials of size [L+1,L+1]

2. Derivatives of Normalized Associated Legendre Polynomials of size [L+1,L+1]

Normalization Constant: https://latex.codecogs.com/png.latex?%5CLARGE%20%5Chat%7BP%7D_%7BN%7D%5E%7B%7CM%7C%7D%28%5Ccos%5Ctheta%29%3D%5Csqrt%7B%5Cfrac%7B2N+1%7D%7B2%7D%20%5Cfrac%7B%28N-%7CM%7C%29%21%7D%7B%28N+%7CM%7C%29%21%7D%7D%20P_%7BN%7D%5E%7B%7CM%7C%7D%28%5Ccos%5Ctheta%29

Phase Factor (-1)^M is omitted.

References:

Limpanuparb, T. and Milthorpe, J., 2014. Associated Legendre Polynomials and Spherical Harmonics Computation for Chemistry Applications. arXiv preprint arXiv:1410.1748.

Li, P. and Jiang, L.J., 2012. The far field transformation for the antenna modeling based on spherical electric field measurements. Progress In Electromagnetics Research, 123, pp.243-261.

Wysin, G.M., 2011. Associated legendre functions and dipole transition matrix elements.

Associated_Legendre_Decimal.py is the extended precision version of Associated_Legendre.py. While it is more accurate, in can be significantly slower compared to floating point precision arithmetic. Also it requires more memory, so it is possible that your computer will run out of memory for L > 400:

Timing Comparison (in seconds):

L = 30 : 0.2833747863769531 | 1.2349169254302979

L = 70 : 1.4793822765350342 | 5.806833505630493

L = 100 : 2.87738299369812 | 11.526304721832275

L = 200 : 12.345782041549683 | 46.637011766433716

L = 300 : 25.131556034088135 | 103.22485566139221

L = 1000 : 302.46400713920593 | ????

Test machine: Intel(R) Core(TM) i7-2620M CPU @ 2.70GHz 4096 KB 4 CPUs 16Gb RAM