Guaranteed Precision for Transcendental and Algebraic Computation Made Easy
Candidate: Zilin Du
Advisor: Chee Keng Yap

Abstract

Numerical non-robustness is a well-known phenomenon when implementing geometric algorithms. A general approach to achieve geometric robustness is Exact Geometric Computation (EGC). This dissertation explores the redesign and extension of Core Library, a C++ library which embraces the EGC approach. The contributions of this thesis are organized into three parts.

In the first part, we discuss the redesign of Core Library, especially the expression "Expr" and bigfloat "BigFloat" classes. Our new design emphasizes extensibility in a clean and modular way. The three facilities in "Expr", filter, root bound and bigfloat, are separated into independent modules. This allows new filters, root bounds and some bigfloat substitute to be plugged in. The key approximate evaluation and precision propagation algorithms have been greatly improved. A new bigfloat system based on MPFR and interval arithmetic has been incorporated. Our benchmark shows that the redesigned Core Library typically has 5-10 times speedup. We also provide tools to facilitate extensions of "Expr" to incorporate new type of nodes, especially transcendental nodes.

Although the Core Library was originally designed for algebraic applications, transcendental functions are needed in many applications. In the second part, we present a complete algorithm for absolute approximation of the general hypergeometric functions. It's complexity is also given. The extension of this algorithm to blackbox number'' is provided. A general hypergeometric function package based on our algorithm is implemented and integrated into the Core Library based on our new design.

Brent has shown that many elementary functions, such as $\exp, \log, \sin$, etc., can be efficiently computed using the Arithmetic-Geometric Mean (AGM) based algorithm. However, he only gave an asymptotic error analysis. The constants in the Big $O(\cdot)$ notation required for implementation are unknown. We provide a non-asymptotic error analysis of the AGM algorithm and the related algorithms for logarithm and exponential functions. These algorithms have been implemented and incorporated into the Core Library.