On Design and Implementation of a Generic Number Type for Real Algebraic Number Computations Based on Expression Dags

Abstract

We report on the design and implementation of a number type called Real_algebraic. This number type allows us to compute the sign of arithmetic expressions involving the operations +, −, ·, /, d √ . The sign computation is always correct and, in this sense, not subject to rounding errors. We focus on modularity and use generic programming techniques to make key parts of the implementation exchangeable. Thus, our design allows for easily performing experiments with different implementations or thereby tailoring the number type for specific tasks. For many problems in computational geometry, instantiations of our number type Real_algebraic are a user-friendly alternative for implementing the exact geometric computation paradigm in order to abandon numerical robustness problems.

Internal Versus External Balancing in the Evaluation of Graph-Based Number Types

“…We present experiments to underline differences between restructuring (Section 2.1) and error bound balancing (Section 2.2). For the comparison, the policy-based exact-decisions number type Real_algebraic with multithreading is used [8,12]. We compare several different strategies.…”

“…This is especially prevalent in the field of computational geometry, where real number computations and combinatorical properties intertwine [10]. In consequence, various exact number types have been developed [6,8,16]. It is an ongoing challenge to make these number types sufficiently efficient to be an acceptable alternative to floating-point primitives in practical applications.…”

Balancing Expression Dags for More Efficient Lazy Adaptive Evaluation

“…We provide a theoretical analysis and evaluate our strategy based on the number type Real algebraic introduced by Mörig et al [6].…”

“…Several expression-dagbased number types have been developed with different evaluation strategies. Strategies can be to gradually increase the precision bottom-up (LEA [1]) or fall back to exact computation (CGAL::Lazy exact nt [8]) if a decision cannot be verified, or to use a precision-driven 1 evaluation (leda::real [2], Core::Expr [5,11], Real algebraic [6]). All of the mentioned number types suffer from high performance overhead compared to standard floating-point arithmetic.…”

“…ImplementationThe balancing strategy has been implemented for the dag-based exact-decision numbertype Real algebraic designed by Mörig et al[6]. This number type consists of a single-or multi-layer floating-point-filter[4], which falls back to adaptive evaluation with bigfloats stored in an expression dag[3].…”