An algorithm for the numerical evaluation of the associated Legendre functions that runs in time independent of degree and order
An algorithm for the numerical evaluation of the associated Legendre functions that runs in time independent of degree and order
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DOI：
10.1016/j.jcp.2018.01.014
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发表时间：
2017-07
期刊：
影响因子：
--
通讯作者：
J. Bremer
中科院分区：
文献类型：
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作者：
J. Bremer
我们描述了在区间（- 1,1）内参数的阶数为0≤ν≤1,000,000，阶数为- ν≤μ≤ν的相关Legendre函数P ν−μ和Q ν−μ的归一化版本的数值求值方法。我们的算法与ν和μ无关，它是基于这样一个事实：虽然相关的勒让德函数本身通过多项式展开表示是非常昂贵的，但定义它们的微分方程的某些解的对数却不是。我们通过数值预计算精心选择的相关Legendre微分方程解的对数，并通过分段三元Chebyshev展开式表示它们来利用这一点。这些预先计算的展开式，允许在上面提到的大范围参数域上快速评估相关的勒让德函数，并补充了渐近展开式和级数展开式，以便完全覆盖它。数值实验的结果证明了我们的方法的有效性，我们的代码评估相关的勒让德函数是公开的。
We describe a method for the numerical evaluation of normalized versions of the associated Legendre functions P ν− μ and Q ν− μ of degrees 0≤ ν≤ 1, 000, 000 and orders− ν≤ μ≤ ν for arguments in the interval (− 1, 1). Our algorithm, which runs in time independent of ν and μ, is based on the fact that while the associated Legendre functions themselves are extremely expensive to represent via polynomial expansions, the logarithms of certain solutions of the differential equation defining them are not. We exploit this by numerically precomputing the logarithms of carefully chosen solutions of the associated Legendre differential equation and representing them via piecewise trivariate Chebyshev expansions. These precomputed expansions, which allow for the rapid evaluation of the associated Legendre functions over a large swath of parameter domain mentioned above, are supplemented with asymptotic and series expansions in order to cover it entirely. The results of numerical experiments demonstrating the efficacy of our approach are presented, and our code for evaluating the associated Legendre functions is publicly available.