单位球面的Gauß求积
Gauß Cubature for the Surface of the Unit Sphere
AI总结：
本文针对三维单位球面，研究Gauß求积规则的构造，指出高维矩方程求解困难，并总结已获得次数至17的良好规则。
摘要：
Gauß求积（多维数值积分）规则是1维Gauß规则的自然推广。它们在特定点数（函数求值次数）下能精确积分尽可能高次的多项式，在此意义下是最优的。对于光滑被积函数，它们是精确且计算高效的公式。\n 构造Gauß规则的点与权重需要求解矩方程组。在1维情形下，该方程组可转化为线性方程组，并获得唯一解，其点位于积分区域内，权重均为正。这些性质有助于保证数值稳定性，我们将此类规则称为“良好”的。在多维情形下，矩方程为非线性代数方程，解甚至不一定存在，更不用说“良好”了。方程组的大小和次数随所需求积规则的次数增加而增长。当待解多项式方程的次数超过4时，解析求解通常变得不可能，需要数值近似。解的存在性不确定，加之方程组的大小和次数，使得该问题对数值方法而言十分棘手。\n 对于（完全对称的）$n$维区域，Gauß规则的构造可容易地特化到$U_3$（三维单位球面）的情形。尽管存在上述问题，对于次数不超过17的情况，已构造/发现了$U_3$上的良好Gauß规则。
英文摘要：
Gauß cubature (multidimensional numerical integration) rules are the natural generalisation of the 1D Gauß rules. They are optimal in the sense that they exactly integrate polynomials of as high a degree as possible for a particular number of points (function evaluations). For smooth integrands, they are accurate, computationally efficient formulae. The construction of the points and weights of a Gauß rule requires the solution of a system of moment equations. In 1D, this system can be converted to a linear system, and a unique solution is obtained, for which the points lie within the region of integration, and the weights are all positive. These properties help ensure numerical stability, and we describe the rules as `good'. In the multidimensional case, the moment equations are nonlinear algebraic equations, and a solution is not guaranteed to even exist, let alone be good. The size and degree of the system grow with the degree of the desired cubature rule. Analytic solution generally becomes impossible as the degree of the polynomial equations to be solved goes beyond 4, and numerical approximations are required. The uncertainty of the existence of solutions, coupled with the size and degree of the system makes the problem daunting for numerical methods. The construction of Gauß rules for (fully symmetric) $n$-dimensional regions is easily specialised to the case of $U_3$, the unit sphere in 3D. Despite the problems described above, for degrees up to 17, good Gauß rules for $U_3$ have been constructed/discovered.
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