Nathalie Revol

Motivations for an arbitrary precision interval arithmetic and the MPFI library

This paper justifies why an arbitrary precision interval arithmetic is needed. To provide accurate results, interval computations require small input intervals; this explains why bisection is so often employed in interval algorithms. The MPFI library has been built in order to fulfill this need. Indeed, no existing library met the required specifications. The main features of this library are briefly given and a comparison with a fixed-precision interval arithmetic, on a specific problem, is presented. It shows that the overhead due to the multiple precision is completely acceptable. Eventually, some applications based on MPFI are given: robotics, isolation of polynomial real roots (by an algorithm combining symbolic and numerical computations) and approximation of real roots with arbitrary accuracy.

Taylor models and floating-point arithmetic: Proof that arithmetic operations are validated in COSY

The goal of this paper is to prove that the implementation of Taylor models in COSY, based on floating-point arithmetic, computes results satisfyin- g the «containment property», i.e. guaranteed results. First, Taylor models are defined and their implementation in the COSY software by Makino and Berz is detailed. Afterwards IEEE-754 floating-point arithmetic is introduced. Then the core of this paper is given: the algorithms implemented in COSY for multiplying a Taylor model by a scalar, for adding or multiplying two Taylor models are given and are proven to return Taylor models satisfying the containment property.

A new range-reduction algorithm

Range reduction is a key point for getting accurate elementary function routines. We introduce a new algorithm that is fast for input arguments belonging to the most common domains, yet accurate over the full double precision range.

Multiple Precision Interval Packages: Comparing Different Approaches

We give a survey on packages for multiple precision interval arithmetic, with the main focus on three specific packages. One is a Maple package, intpakX, and two are C/C++ libraries, GMP-XSC and MPFI. We discuss their different features, present timing results and show several applications from various fields, where high precision intervals are fundamental.

Interval Newton Iteration in Multiple Precision for the Univariate Case

In this paper, interval arithmetic using an underlying multiple precision arithmetic is briefly presented. Then interval Newton iteration for solving nonlinear equations is introduced. A new Newtons algorithm based on multiple precision interval arithmetic is given, along with its properties: termination, arbitrary accuracy on the computed zeros, automatic and dynamic adaptation of the precision. Finally, some experiments illustrate the behaviour of this method.

Proposal for a Standardization of Mathematical Function Implementation in Floating-Point Arithmetic

Some aspects of what a standard for the implementation of the mathematical functions could be are presented. Firstly, the need for such a standard is motivated. Then the proposed standard is given. The question of roundings constitutes an important part of this paper: three levels are proposed, ranging from a level relatively easy to attain (with fixed maximal relative error) up to the best quality one, with correct rounding on the whole range of every function. We do not claim that we always suggest the right choices, or that we have thought about all relevant issues. The mere goal of this paper is to raise questions and to launch the discussion towards a standard.

Numerical Reproducibility and Parallel Computations: Issues for Interval Algorithms

What is called numerical reproducibility is the problem of getting the same result when the scientific computation is run several times, either on the same machine or on different machines, with different types and numbers of processing units, execution environments, computational loads, etc. This problem is especially stringent for HPC numerical simulations. In what follows, we identify the problems encountered when implementing interval routines in floating-point arithmetic. Some are well-known and common in numerical computations, some are specific to interval computations. We propose here a classification of floating-point issues by distinguishing their severity with respect to correctness and tightness of the computed interval result. In fact, interval computation can accommodate the lack of numerical reproducibility as long as it does not affect the inclusion property, which is the main property of interval arithmetic. Several ways to preserve the inclusion property are presented, on the example of the product of matrices with interval coefficients.

A Validated Real Function Calculus

We present a framework for validated numerical computations with real functions. The framework is based on a formalisation of abstract data types for basic floating-point arithmetic, interval arithmetic and function models based on Banach algebra. As a concrete instantiation, we develop an elementary smooth function calculus approximated by sparse polynomial models. We demonstrate formal verification applied to validated calculus by a formalisation of basic arithmetic operations in a theorem prover. The ultimate aim is to develop a formalism powerful enough for reachability analysis of nonlinear hybrid systems.

Standardized interval arithmetic and interval arithmetic used in libraries

The standardization of interval arithmetic is currently undertaken by the IEEE-1788 working group. Some features of the standard are detailed. The features chosen here are the ones which may be the less widely adopted in current implementations of interval arithmetic. A survey of interval-based libraries, focusing on these features, is given.

A new range reduction algorithm

Range reduction is a key point for getting accurate elementary function routines. We introduce a new algorithm that is fast for input arguments belonging to the most common domains, yet accurate over the full double precision range.