Gaussian quadratures over the surface of the sphere

Jörg Waldvogel

Seminar für angewandte Mathematik, ETH, CH-8092 Zürich, Switzerland

waldvogel@sam.math.ethz.ch

Contributed talk

The computation of integrals $I := \int_{S_2} f(x) \,d\omega$ over the surface $S_2$ of the unit sphere in $\bf {R}^3$ is an important practical task (needed for, e.g., accumulating the radiation influx from all directions of space). However, such integrations often suffer from the nonexistence of regular parametrizations of $S_2$. Therefore, numerical approximations

$$I = \sum_{k=1}^N w_k \,f(x_k) + R_N$$

with points

$$x_k = (x_k^1, x_k^2, x_k^3) \in S_2, ~~~\sum_{j=1}^3 (x_k^j)^2 = 1$$

and weights $w_k, ~~k=1,\dots,N$ and a remainder $R_N$ are of particular interest. As in one-dimensional Gaussian quadrature, $x_k, w_k$ are chosen such that $R_N = 0$ for all polynomials f in $x^1, x^2, x^3$ of total degree $\le D$, where D is as large as possible.

The icosahedral symmetry of the set of points is of particular interest; many formulas of high precision are expected to exist. We use the monomials in 2 invariant polynomials (of degrees 6 and 10) in order to generate the space of all polynomials on the sphere. The resulting systems of nonlinear equations are solved numerically. The classical formulas of orders $D \leq 15$ with $N \leq 120$ points are easily recovered. A larger example is $D = 31, N = 450$, and many more examples were constructed.