生成多元正交多项式的Stieltjes算法
A Stieltjes algorithm for generating multivariate orthogonal polynomials
摘要：
多元的正交多项式具有向量值三项递归关系，很像对应的一维关系。这种关系只需要某些递归矩阵的知识，并允许简单和稳定的多元正交多项式的评估。在单变量的情况下，各种算法可以计算递归系数的能力，计算多项式的时刻，但这样的程序是不存在的多维。我们提出了一个新的多元Stieltjes（MS）算法，填补了这一空白，在多元的情况下，允许计算递归矩阵假设时刻可用。该算法基本上是明确的二维和三维，但需要数值解的非凸问题在超过三维。直接Gram-Schmidt型正交化相比，我们证明了在几个例子中，在多达三个维度的MS算法是更稳定，并允许精确计算的正交基在多变量设置，在对比直接正交化方法。
英文摘要：
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.
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