Finance

Wavenumber

Physics

Legendre function

Mathematics

Term (time)

Evolutionary biology

Biology

Algorithm

Optics

Diagonal

Legendre polynomials

Mathematical analysis

Function (biology)

Series (stratigraphy)

Trigonometric functions

Paleontology

Order (exchange)

Economics

Acoustics

Degree (music)

Sine

Geometry

Spherical harmonics

Computation

Quantum mechanics

Applied mathematics

Journal of Geodetic Science

In order to accelerate the spherical/spheroidal harmonic synthesis of any function, we developed a new recursive method to compute the sine/cosine series coefficient of the 4π fully- and Schmidt quasi-normalized associated Legendre functions. The key of the method is a set of increasing-degree/order mixed-wavenumber two to four-term recurrence formulas to compute the diagonal terms. They are used in preparing the seed values of the decreasing-order fixed-degree, and fixed-wavenumber two- and three-term recurrence formulas, which are obtained by modifying the classic relations. The new method is accurate and capable to deal with an arbitrary high degree/ order/wavenumber. Also, it runs significantly faster than the previous method of ours utilizing the Wigner d function, say around 20 times more when the maximum degree exceeds 1,000.

Fast computation of sine/cosine series coefficients of associated Legendre function of arbitrary high degree and order