Evaluation of toroidal harmonics Original Research Article

Three algorithms to evaluate toroidal harmonics, i.e., Legendre functions of integral order and half-odd degree of the first and second kinds for real arguments larger than one, are presented. The first algorithm (DTORH1) allows the evaluation of the set {Pn−1/2m(x),Qn−1/2m(x)} for fixed (integer and positive) values of m and n=0,1,…,N. The algorithms DTORH2 and DTORH3 extend the method used in DTORH1 to obtain the set {Pn−1/2m(x),Qn−1/2m(x)} for m=0,…,M and n=0,1,…,N. The output of DTORH2 is equivalent to the result of the application of DTORH1 M times (m=0,1,…,M). However, due to the organization of the algorithm, DTORH2 is faster than DTORH1 when several orders and degrees are calculated. DTORH2 is better suited than DTORH3 when high orders and degrees are needed. On the other hand DTORH3, though more restrictive on the maximum evaluable degrees N, is even faster than DTORH2. Our tests of accuracy, show that the three codes achieve a precision of one part in 1012. We discuss the performance of our codes, their speed and their range of validity. The application of the algorithms to solve Laplaceʹs and Poissonʹs equations in toroidal coordinates is discussed and an explicit numerical example is shown.