A new Fortran 90 program to compute regular and irregular associated Legendre functions

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• 作者：Barry I. Schneider ; Javier Segura ; Amparo Gil ; Xiaoxu Guan ; Klaus Bartschat
• 关键词：Associated Legendre functions ; Recursion relations ; Miller's algorithm ; Continued fractions
• 刊名：Computer Physics Communications
• 出版年：2010
• 出版时间：December 2010
• 年：2010
• 卷：181
• 期：12
• 页码：2091-2097
• 全文大小：174 K

文摘

We present a modern Fortran 90 code to compute the regular and irregular associated Legendre functions for all x(−1,+1) (on the cut) and x>1 and integer degree (l) and order (m). The code applies either forward or backward recursion in (l) and (m) in the stable direction, starting with analytically known values for forward recursion and considering both a Wronskian based and a modified Miller's method for backward recursion. While some Fortran 77 codes existed for computing the functions off the cut, no Fortran 90 code was available for accurately computing the functions for all real values of x different from x=±1 where the irregular functions are not defined.

Program summary

Program title: Associated Legendre Functions

Catalogue identifier: AEHE_v1_0

Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEHE_v1_0.html

Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland

Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html

No. of lines in distributed program, including test data, etc.: 6722

No. of bytes in distributed program, including test data, etc.: 310 210

Distribution format: tar.gz

Programming language: Fortran 90

Computer: Linux systems

Operating system: Linux

RAM: bytes

Classification: 4.7

Nature of problem: Compute the regular and irregular associated Legendre functions for integer values of the degree and order and for all real arguments. The computation of the interaction of two electrons, 1/r₁−r₂, in prolate spheroidal coordinates is used as one example where these functions are required for all values of the argument and we are able to easily compare the series expansion in associated Legendre functions and the exact value.

Solution method: The code evaluates the regular and irregular associated Legendre functions using forward recursion when x<1 starting the recursion with the analytically known values of the first two members of the sequence. For values of the argument x<1, the upward recursion over the degree for the regular functions is numerically stable. For the irregular functions, backward recursion must be applied and a suitable method of starting the recursion is required. The program has two options; a modified version of Miller's algorithm and the use of the Wronskian relation between the regular and irregular functions, which was the method considered in [1]. Both approaches require the computation of a continued fraction to begin the recursion. The Wronskian method (which can also be described as a modified Miller's method) is a convenient method of computations when both the regular and irregular functions are needed.

Running time: The example tests provided take a few seconds to run.

References:

[1] A. Gil, J. Segura, A code to evaluate prolate and oblate spheroidal harmonics, Comput. Phys. Commun. 108 (1998) 267–278.