Mathematics – Numerical Analysis
Scientific paper
2000-03-03
Mathematics
Numerical Analysis
Unpublished BScHons thesis in the Department of Mathematics at The University of Queensland, Australia. Thesis submitted Novem
Scientific paper
Gau{\ss} cubature (multidimensional numerical integration) rules are the natural generalisation of the 1D Gau{\ss} 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 Gau{\ss} 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 Gau{\ss} 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 Gau{\ss} rules for $U_3$ have been constructed/discovered.
Wit David de
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