Norbert Th. Müller

The iRRAM: Exact Arithmetic in C++

The iRRAM is a very efficient C++ package for error-free real arithmetic based on the concept of a Real-RAM. Its capabilities range from ordinary arithmetic over trigonometric functions to linear algebra even with sparse matrices. We discuss the concepts and some highlights of the implementation.

Novel Approaches to Numerical Software with Result Verification

Traditional design of numerical software with result verification is based on the assumption that we know the algorithm f(x 1,...,x n ) that transforms inputs x 1,...,x n into the output y=f(x 1,...,x n ), and we know the intervals of possible values of the inputs. Many real-life problems go beyond this paradigm. In some cases, we do not have an algorithm f, we only know some relation (constraints) between x i and y. In other cases, in addition to knowing the intervals x i , we may know some relations between x i ; we may have some information about the probabilities of different values of x i , and we may know the exact values of some of the inputs (e.g., we may know that x 1 = π/2). In this paper, we describe the approaches for solving these real-life problems. In Section 2, we describe interval consistency techniques related to handling constraints; in Section 3, we describe techniques that take probabilistic information into consideration, and in Section 4, we overview techniques for processing exact real numbers.

Subpolynomial Complexity Classes of Real Functions and Real Numbers

In this paper a definition of computability and complexity of real functions and real numbers is given which is open to methods of recursive function theory as well as to methods of numerical analysis. As an example of application we study the computational complexity of roots and thereby establish a subpolynomial hierarchy of real closed fields.

Computability on random variables

Abstract In this paper, we study aspects of computability concerning random variables under the background of Type 2 Theory of Effectivity (TTE). We show that the resulting definitions are “natural” as they suffice to successfully discuss questions of computability of basic queueing systems, e.g. of the so-called M/G/1-system.

Complexity of Operators on Compact Sets

Based on oracle Turing machines, we investigate the computational complexity of operators on compact sets. For the projection and convex hull we are able to show exponential upper and lower bounds as well as a connection to the P=NP problem for special settings.

Making big steps in trajectories

We consider the solution of initial value problems within the context of hybrid systems and emphasise use of high precision approximations (in software for exact real arithmetic). We propose a novel algorithm for the computation of trajectories up to the area where discontinuous jumps appear, applicable for holomorphic flow functions. Examples with a prototypical implementation illustrate that the algorithm might provide results with higher precision than well-known ODE solvers at a similar computation time.

Jordan Areas and Grids

Jordan curves can be used to represent special subsets of the Euclidean plane, either the (open) interior of the curve or the (compact) union of the interior and the curve itself. We compare the latter with other representations of compact sets using grids of points and we are able to show that knowing the length of a rectifiable curve is sufficient to translate from the grid representation to the Jordan curve.

Software Numerical Instability Detection and Diagnosis by Combining Stochastic and Infinite-Precision Testing

Numerical instability is a well-known problem that may cause serious runtime failures. This paper discusses the reason of instability in software development process, and presents a toolchain that not only detects the potential instability in software, but also diagnoses the reason for such instability. We classify the reason of instability into two categories. When it is introduced by software requirements, we call the instability caused by problem . In this case, it cannot be avoided by improving software development, but requires inspecting the requirements, especially the underlying mathematical properties. Otherwise, we call the instability caused by practice. We design our toolchain as four loosely-coupled tools, which combine stochastic arithmetic with infinite-precision testing. Each tool in our toolchain can be configured with different strategies according to the properties of the analyzed software. We evaluate our toolchain on subjects from literature. The results show that it effectively detects and separates the instabilities caused by problems from others. We also conduct an evaluation on the latest version of GNU Scientific Library, and the toolchain finds a few real bugs in the well-maintained and widely deployed numerical library. With the help of our toolchain, we report the details and fixing advices to the GSL buglist.

From Calculus to Algorithms without Errors

Using mathematics within computer software almost always includes the necessity to compute with real (or complex) numbers. However, implementations often just use the 64-bit double precision data type. This may lead to serious stability problems even for mathematically correct algorithms. There are many ways to reduce these software-induced stability problems, for example quadruple or multiple-precision data types, interval arithmetic, or even symbolic computation. We propagate Exact Real Arithmetic (ERA) as a both convenient and practically efficient framework for rigorous numerical algorithms.

Towards Using Exact Real Arithmetic for Initial Value Problems

In the paper we report on recent developments of the iRRAM software [7] for exact real computations. We incorporate novel methods and tools to generate solutions of initial value problems for ODE systems with polynomial right hand sides (PIVP). The algorithm allows the evaluation of the solutions with an arbitrary precision on their complete open intervals of existence. In consequence, the set of operators implemented in the iRRAM software (like function composition, computation of limits, or evaluation of Taylor series) is expanded by PIVP solving.