基于区间算术的有效拓扑度计算
Effective Topological Degree Computation Based on Interval Arithmetic
摘要：
本文描述了一种计算拓扑度deg（f，B，0）的新算法，其中B \subseteq Rn是闭实区间的乘积，f：B \rightarrow Rn是以算术表达式形式给出的实值连续函数.该算法将数值计算与组合计算完全分离。基于此，数值部分仅可证明地计算对于随后的组合部分严格必要的信息，并且组合部分可以基于之前计算的数值信息来优化其计算。我们还提出了计算实验的基础上实现的算法。此外，与以前的工作相比，该算法不假设知识的Lipschitz常数的函数f，并适用于任意连续函数的区间算术的一些概念可以定义。
英文摘要：
We describe a new algorithm for calculating the topological degree deg (f, B, 0) where B \subseteq Rn is a product of closed real intervals and f : B \rightarrow Rn is a real-valued continuous function given in the form of arithmetical expressions. The algorithm cleanly separates numerical from combinatorial computation. Based on this, the numerical part provably computes only the information that is strictly necessary for the following combinatorial part, and the combinatorial part may optimize its computation based on the numerical information computed before. We also present computational experiments based on an implementation of the algorithm. Also, in contrast to previous work, the algorithm does not assume knowledge of a Lipschitz constant of the function f, and works for arbitrary continuous functions for which some notion of interval arithmetic can be defined.
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