论文标题
用于生成多元正交多项式的Stieltjes算法
A Stieltjes algorithm for generating multivariate orthogonal polynomials
论文作者
论文摘要
几个变量的正交多项式具有矢量值的三项复发关系，就像相应的一维关系。这种关系只需要了解某些复发矩阵，并允许对多元正交多项式进行简单稳定的评估。在单变量的情况下，鉴于能够计算多项式矩的能力，各种算法可以评估复发系数，但是在多个维度中不存在这样的过程。我们提出了一种新的多元stieltjes（MS）算法，该算法填补了在多元式情况下的空白，从而允许计算复发矩阵，假设有矩可用。该算法在两个维度和三个维度上基本上是显式的，但是需要在超过三个维度中针对非凸问题的数值解决方案。与直接的革兰氏型正交型正交化相比，我们在多达三个维度的几个示例上证明了MS算法要稳定得多，并且可以准确地计算多变量设置中的正交基础，这与直接正交方法相反。
Orthogonal polynomials of several variables have a vector-valued three-term recurrence relation, much like the corresponding one-dimensional relation. This relation requires only knowledge of certain recurrence matrices, and allows simple and stable evaluation of multivariate orthogonal polynomials. In the univariate case, various algorithms can evaluate the recurrence coefficients given the ability to compute polynomial moments, but such a procedure is absent in multiple dimensions. We present a new Multivariate Stieltjes (MS) algorithm that fills this gap in the multivariate case, allowing computation of recurrence matrices assuming moments are available. The algorithm is essentially explicit in two and three dimensions, but requires the numerical solution to a non-convex problem in more than three dimensions. Compared to direct Gram-Schmidt-type orthogonalization, we demonstrate on several examples in up to three dimensions that the MS algorithm is far more stable, and allows accurate computation of orthogonal bases in the multivariate setting, in contrast to direct orthogonalization approaches.
