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.
翻译:多个变量的正方形多数值复发关系具有矢量价值的三期重现关系, 类似于相应的一维关系 。 此关系只要求了解某些复发矩阵, 并允许对多变量或正方形多圆形矩阵进行简单稳定的评估 。 在单方形情况下, 各种算法可以评估复发系数, 因为它能够计算多数值时段, 但这样的程序在多个维度中不存在 。 我们提出了一个新的多变量Stieltjes( MS) 算法, 以填补多变量情况下的这一缺口, 允许计算复发矩阵时段。 该算法基本上在两个和三个维度上是明确的, 但需要用数字方法解决一个超过三个维度的非二次曲线问题。 与直接的 Gram- Smidt 类型或otocalization 相比, 我们用三个维的示例来证明, MS 算法非常稳定得多, 并且可以精确计算多变量设置中的正方形基数基值, 与直接的或正态方法不同 。