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

Abstract We present a modern Fortran 90 code to compute the regular Plm(x) and irregular Qlm(x) 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 t... Title of program: Associated Legendre Functions Catalogue Id: AEHE_v1_0 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. Versions of this program held in the CPC repository in Mendeley Data AEHE_v1_0; Associated Legendre Functions; 10.1016/j.cpc.2010.08.038 This program has been imported from the CPC Program Library held at Queen's University Belfast (1969-2018)摘要 本文提出一款现代化的Fortran 90代码，用于计算正则连带勒让德函数（Associated Legendre Functions）P_l^m(x)与非正则连带勒让德函数Q_l^m(x)，其中x可覆盖区间(-1,+1)（割线上）以及|x|>1的全体实数，阶数l与次数m均为整数。该代码可沿稳定方向分别对阶数l与次数m实施正向或反向递推：正向递推以解析已知的初始值为起点，反向递推则同时考虑基于朗斯基行列式（Wronskian）的方法与改进型Miller算法（Miller's method）。此前虽已有部分Fortran 77代码可用于计算…… 程序名称：连带勒让德函数 馆藏编号：AEHE_v1_0 问题属性 针对整数阶数与次数的连带勒让德函数，以及全体实自变量完成计算。以长椭球坐标系（prolate spheroidal coordinates）下两电子相互作用项1/|r₁ - r₂|的计算为例，该场景需要覆盖全部自变量取值范围的连带勒让德函数，我们可借此便捷对比连带勒让德函数的级数展开结果与精确解析值。 本程序存放在Mendeley数据的CPC库内的版本信息：AEHE_v1_0；连带勒让德函数；10.1016/j.cpc.2010.08.038 本程序源自贝尔法斯特女王大学馆藏的CPC程序库（1969-2018）