A new recurrence formula for efficient computation of spherical harmonic transform

Abstract

A new recurrence formula to calculate the associated Legendre functions is proposed for efficient computation of the spherical harmonic transform. This new recurrence formula makes the best use of the fused multiply-add (FMA) operations implemented in modern computers. The computational speeds in calculating the spherical harmonic transform are compared between a numerical code in which the new recurrence formula is implemented and another code using the traditional recurrence formula. This comparison shows that implementation of the new recurrence formula contributes to a faster transform. Furthermore, a scheme to maintain the accuracy of the transform, even when the truncation wavenumber is huge, is also explained.