Legendre.jlseries contains a detailed article considering the numerical accuracy of the recurrence implementation as a whole. This short addendum presents a mathematical transformation which is used to improve the accuracy of one of the recurrence relation coefficients for the spherical harmonic normalization.
Numerically computing any non-trivial function (spanning the trigonometric functions through to special functions like the complex gamma function) is a large field of numerical computing, and one I am interested in expanding my knowledge within. In this article, I describe my recent exploration of how the exponential function can be implemented numerically.
An algorithm for calculating special functions is no use if it has poor numerical accuracy and returns inaccurate or invalid answers. In this article I discuss the importance of using fused multiply-add (FMA) and a form of loop-unrolling to maintain numerical accuracy in calculating the associated Legendre polynomials to degrees and orders of order 1000 and greater.