Results 1 
9 of
9
Motivations for an arbitrary precision interval arithmetic and the MPFI library
 Reliable Computing
, 2002
"... Nowadays, computations involve more and more operations and consequently errors. The limits of applicability of some numerical algorithms are now reached: for instance the theoretical stability of a dense matrix factorization (LU or QR) is ensured under the assumption that n 3 u < 1, where n is the ..."
Abstract

Cited by 29 (7 self)
 Add to MetaCart
Nowadays, computations involve more and more operations and consequently errors. The limits of applicability of some numerical algorithms are now reached: for instance the theoretical stability of a dense matrix factorization (LU or QR) is ensured under the assumption that n 3 u < 1, where n is the dimension of the matrix and u = 1 + − 1, with 1 + the smallest floatingpoint larger than 1; this means that n must be less than 200,000, which is almost reached by modern simulations. The numerical quality of solvers is now an issue, and not only their mathematical quality. Let us cite studies performed by the CEA (French Nuclear Agency) on the simulation of nuclear plant accidents and also softwares controlling and possibly correcting numerical programs, such as Cadna [10] or Cena [20]. Another approach consists in computing with certified enclosures, namely interval arithmetic [21, 2, 18]. The fundamental principle of this arithmetic consists in replacing every number by an interval enclosing it. For instance, π cannot be exactly represented using a binary or decimal arithmetic, but it
Theory of real computation according to EGC
 In Proceedings of the Dagstuhl Seminar on Reliable Implementation of Real Number Algorithms: Theory and Practice, Lecture Notes in Computer Science
, 2006
"... The Exact Geometric Computation (EGC) mode of computation has been developed over the last decade in response to the widespread problem of numerical nonrobustness in geometric algorithms. Its technology has been encoded in libraries such as LEDA, CGAL and Core Library. The key feature of EGC is the ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
The Exact Geometric Computation (EGC) mode of computation has been developed over the last decade in response to the widespread problem of numerical nonrobustness in geometric algorithms. Its technology has been encoded in libraries such as LEDA, CGAL and Core Library. The key feature of EGC is the necessity to decide zero in its computation. This paper addresses the problem of providing a foundation for the EGC mode of computation. This requires a theory of real computation that properly addresses the Zero Problem. The two current approaches to real computation are represented by the analytic school and algebraic school. We propose a variant of the analytic approach based on real approximation. • To capture the issues of representation, we begin with a reworking of van der Waerden’s idea of explicit rings and fields. We introduce explicit sets and explicit algebraic structures. • Explicit rings serve as the foundation for real approximation: our starting point here is not R, but F ⊆ R, an explicit ordered ring extension of Z that is dense in R. We develop the approximability of real functions within standard Turing machine computability, and show its connection to the analytic approach. • Current discussions of real computation fail to address issues at the intersection of continuous and discrete computation. An appropriate computational model for this purpose is obtained by extending Schönhage’s pointer machines to support both algebraic and numerical computation. • Finally, we propose a synthesis wherein both the algebraic and the analytic models coexist to play complementary roles. Many fundamental questions can now be posed in this setting, including transfer theorems connecting algebraic computability with approximability. 1
RZ: A tool for bringing constructive and computable mathematics closer to programming practice
 CiE 2007: Computation and Logic in the Real World, volume 4497 of LNCS
, 2007
"... Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Obje ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Objective Caml. The system supports a rich input language allowing descriptions of complex mathematical structures. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools. 1
Guaranteed Precision for Transcendental and Algebraic Computation made Easy
, 2006
"... Dedicated to the friends and families who blessed and supported me iv ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Dedicated to the friends and families who blessed and supported me iv
On the Stability of Fast Polynomial Arithmetic
"... Operations on univariate dense polynomials—multiplication, division with remainder, multipoint evaluation—constitute central primitives entering as buildup blocks into many higher applications and algorithms. Fast Fourier Transform permits to accelerate them from naive quadratic to running time O(n ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Operations on univariate dense polynomials—multiplication, division with remainder, multipoint evaluation—constitute central primitives entering as buildup blocks into many higher applications and algorithms. Fast Fourier Transform permits to accelerate them from naive quadratic to running time O(n · polylogn), that is softly linear in the degree n of the input. This is routinely employed in complexity theoretic considerations and, over integers and finite fields, in practical number theoretic calculations. The present work explores the benefit of fast polynomial arithmetic over the field of real numbers where the precision of approximation becomes crucial. To this end, we study the computability of the above operations in the sense of Recursive Analysis as an effective refinement of continuity. This theoretical worstcase stability analysis is then complemented by an empirical evaluation: We use GMP and the iRRAM to find the precision required for the intermediate calculations in order to achieve a desired output accuracy. 1
The Design of Core 2: A Library for Exact Numeric Computation in Geometry and Algebra
 In Third International Congress on Mathematical Software, volume 6327, Kobe, Japon
, 2010
"... Abstract. There is a growing interest in numericalgebraic techniques in the computer algebra community as such techniques can speed up many applications. This paper is concerned with one such approach called Exact Numeric Computation (ENC). The ENC approach to algebraic number computation is based ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. There is a growing interest in numericalgebraic techniques in the computer algebra community as such techniques can speed up many applications. This paper is concerned with one such approach called Exact Numeric Computation (ENC). The ENC approach to algebraic number computation is based on iterative verified approximations, combined with constructive zero bounds. This paper describesCore 2, the latest version of the Core Library, a package designed for applications such as nonlinear computational geometry. The adaptive complexity of ENC combined with filters makes such libraries practical. Core 2 smoothly integrates our algebraic ENC subsystem with transcendental functions with εaccurate comparisons. This paper describes how the design of Core 2 addresses key software issues such as modularity, extensibility, and efficiency in a setting that combines algebraic and transcendental elements. Our redesign preserves the original goals of the Core Library, namely, to provide a simple and natural interface for ENC computation to support rapid prototyping and exploration. We present examples, experimental results, and timings for our new system, released as Core Library 2.0. 1
§1. What is Numerical Nonrobustness? Lecture 1 Page 1 Lecture 1 INTRODUCTION TO NUMERICAL
"... This chapter gives an initial orientation to some key issues that concern us. What is the nonrobustness phenomenon? Why does it appear so intractable? Of course, the prima facie reason for nonrobustness ..."
Abstract
 Add to MetaCart
This chapter gives an initial orientation to some key issues that concern us. What is the nonrobustness phenomenon? Why does it appear so intractable? Of course, the prima facie reason for nonrobustness
www.informatik2011.de The Trouble with Real Numbers (Invited Paper)
"... Abstract: Comprehensive analytical modeling and simulation of cyberphysical systems is an integral part of the process that brings to life novel designs and products. But the effort needed to go from analytical models to running simulation code can impede or derail this process. Our thesis is that ..."
Abstract
 Add to MetaCart
Abstract: Comprehensive analytical modeling and simulation of cyberphysical systems is an integral part of the process that brings to life novel designs and products. But the effort needed to go from analytical models to running simulation code can impede or derail this process. Our thesis is that this process is amenable to automation, and that automating it will accelerate the pace of innovation. This paper reviews some basic concepts that we found interesting or thought provoking, and articulates some questions that may help prove or disprove this thesis. While based on ideas drawn from different disciplines outside programming languages, all these observations and questions pertain to how we need to reason and compute with real numbers. 1