Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2000
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/math/0003018 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908599895195648 |
|---|---|
| author | De Wit, David |
| author_facet | De Wit, David |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0003018 |
| institution | arXiv |
| publishDate | 2000 |
| record_format | arxiv |
| spellingShingle | Gauß Cubature for the Surface of the Unit Sphere De Wit, David Numerical Analysis 65D32 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. |
| title | Gauß Cubature for the Surface of the Unit Sphere |
| topic | Numerical Analysis 65D32 |
| url | https://arxiv.org/abs/math/0003018 |