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Manuscript Title: Evaluation of toroidal harmonics.
Authors: J. Segura, A. Gil
Program title: DTORH1, DTORH2, DTORH3
Catalogue identifier: ADKV_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 124(2000)104
Programming language: Fortran.
Computer: HP Model 715/100, SUN Enterprise 3000, PC Pentium MMX 200MHz.
Operating system: UNIX, Linux.
Keywords: General purpose, Function, Toroidal harmonics, Legendre functions, Laplace's equation, Toroidal coordinates.
Classification: 4.7.

Nature of problem:
We include three codes to evaluate toroidal harmonics:
  • DTORH1: This code evaluates toroidal harmonics (TH) of the first and second kinds for a given order m, from the lowest (positive) degreee (n=0) to a maximum degree n = N in the same run.
  • DTORH2: This code evaluates toroidal harmonics of the first and second kinds Pm(n-1/2)(x) and Qm(n-1/2)(x) for orders m = 0,...,M and degrees n = 0,...,N. In this code, for each given order m, TH up to the maximum degree reached by the computer for such order m (N(m)) can be calculated.
  • DTORH3: This code evaluates toroidal harmonics of the first and second kinds for orders m = 0,...,M and degrees n = 0,...,N. TH up to the smallest maximum degree reachable for the different values of m = 0,1,...,M can be calculated.
The algorithms find their application in problems with toroidal geometry. In particular, the potential problem for a torus is solved using both Ps and Qs.

Solution method:
The codes are based on the application of recurrence relations both over m and n. The recurrence over n is applied forward for the Ps and backward for the Qs while, over m, the situation is the contrary: the Ps are evaluated backwards and the Qs forward.
The forward and backward recursions (over n or over m) are linked through continued fractions for the ratio of minimal solutions and Wronskian relations.
The recurrences are fed with the two starting values Q0(1/2) and Q1(-1/2), which are evaluated using Carlson's duplication formula for elliptic integrals.

Restrictions:
The maximum degree (order) that can be reached with our method, for a given order (degree) m (n) and for a fixed real positive value of x, is provided by the maximum real number defined in our machine. The user can choose two different relative accuracies (10**-8 or 10**-12) in the interval 1.001 < x < 1000 for all available values of the orders and degrees. For more details see text: LONG WRITE-UP, section 3.

Running time:
Depends on the values of the argument x, the orders (m) and the degrees (n). As an example, to evaluate the set {Pm(n-1/2)(1.1), Qm(n-1/2)(1.1), m = 0,...,10, n = 0,...,50,} the code DTORH3 spends 2.6 ms in a HP 715/100.
For more details see text: LONG WRITE-UP, section 5.