Dagstuhl Seminar Proceedings, Volume 6021
Dagstuhl Seminar Proceedings
DagSemProc
https://www.dagstuhl.de/dagpub/1862-4405
https://dblp.org/db/series/dagstuhl
1862-4405
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
6021
2006
https://drops.dagstuhl.de/entities/volume/DagSemProc-volume-6021
06021 Abstracts Collection – Reliable Implementation of Real Number Algorithms: Theory and Practice
From 08.01.06 to 13.01.06, the Dagstuhl Seminar 06021 ``Reliable Implementation of Real Number Algorithms: Theory and Practice'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper.
The first section describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.
Real number algorithms
reliable implementation
1-18
Regular Paper
Peter
Hertling
Peter Hertling
Christoph M.
Hoffmann
Christoph M. Hoffmann
Wolfram
Luther
Wolfram Luther
Nathalie
Revol
Nathalie Revol
10.4230/DagSemProc.06021.1
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06021 Summary – Reliable Implementation of Real Number Algorithms: Theory and Practice
The seminar brought together researchers from many different disciplines concerned with the reliable implementation of real number algorithms either from a theoretical or from a practical point of view. In this summary we describe the topics, the goals, and the contributions of the seminar.
Real number computability
real number algorithms
reliable computing
algorithms with result verification
interval arithmetic
geometric computing
robustness
solid modeling
1-5
Regular Paper
Peter
Hertling
Peter Hertling
Christoph M.
Hoffmann
Christoph M. Hoffmann
Wolfram
Luther
Wolfram Luther
Nathalie
Revol
Nathalie Revol
10.4230/DagSemProc.06021.2
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A Descartes Algorithms for Polynomials with Bit-Stream Coefficients
The Descartes method is an algorithm for isolating the
real roots of square-free polynomials with real coefficients. We assume
that coefficients are given as (potentially infinite) bit-streams. In other
words, coefficients can be approximated to any desired accuracy, but are not
known exactly. We show that
a variant of the Descartes algorithm can cope with bit-stream
coefficients. To isolate the real roots of a
square-free real polynomial $q(x) = q_nx^n+ldots+q_0$ with root
separation $
ho$, coefficients $abs{q_n}ge1$ and $abs{q_i} le 2^ au$,
it needs coefficient approximations to $O(n(log(1/
ho) + au))$
bits after the binary point and has an expected cost of
$O(n^4 (log(1/
ho) + au)^2)$ bit operations.
Root Isolation
Interval Arithmetic
Descartes Algorithm
1-12
Regular Paper
Kurt
Mehlhorn
Kurt Mehlhorn
Arno
Eigenwillig
Arno Eigenwillig
Lutz
Kettner
Lutz Kettner
Werner
Krandick
Werner Krandick
Susanne
Schmitt
Susanne Schmitt
Nicola
Wolpert
Nicola Wolpert
10.4230/DagSemProc.06021.3
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A Proposal to add Interval Arithmetic to the C++ Standard Library
I will report on a recent effort by Guillaume Melquiond, HervÃƒÂ© Br"onnimann
and myself to push forward a proposal to include interval arithmetic in the
next C++ ISO standard. The goals of the standardization are to produce a
unified specification which will serve as many uses of intervals as possible,
together with hoping for very efficient implementations, closer to the
compilers. I will describe how the standardization process works, explain
some of the design choices made, and list some of the other questions arising
in the process. We welcome any comment on the proposal.
Interval arithmetic
C++
ISO standard
1-25
Regular Paper
Sylvain
Pion
Sylvain Pion
Hervé
Brönnimann
Hervé Brönnimann
Guillaume
Melquiond
Guillaume Melquiond
10.4230/DagSemProc.06021.4
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Floating Point Geometric Algorithms for Topologically Correct Scientific Visualization
The unresolved subtleties of floating point computations in geometric modeling become considerably more difficult in animations and scientific visualizations.
Some emerging solutions based upon topological considerations will be presented.
A novel geometric seeding algorithm for Newton's method was used in experiments to determine feasible support for these visualization applications.
Geometry
algorithm
visualization
1-11
Regular Paper
Thomas J.
Peters
Thomas J. Peters
Edward L. F.
Moore
Edward L. F. Moore
10.4230/DagSemProc.06021.5
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Interval Arithmetic Using SSE-2
We present an implementation of double precision interval arithmetic using the single-instruction-multiple-data SSE-2 instruction and register set extensions. The implementation is part of a package for exact real arithmetic, which defines the interval arithmetic variation that must be used: incorrect operations such as division by zero cause exceptions, loose evaluation of the operations is in effect, and performance is more important than tightness of the produced bounds. The SSE2 extensions are suitable for the job, because they can be used to operate on a pair of double precision numbers and include separate rounding mode control and detection of the exceptional conditions. The paper describes the ideas we use to fit interval arithmetic to this set of instructions, shows a performance comparison with other freely available interval arithmetic packages, and discusses possible very simple hardware extensions that can significantly increase the performance of interval arithmetic.
Interval Arithmetic
SSE2
1-12
Regular Paper
Branimir
Lambov
Branimir Lambov
10.4230/DagSemProc.06021.6
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Interval Subroutine Library Mission
We propose the collection, standardization, and distribution of a full-featured production quality library for reliable scientific computing with routines using interval techniques for use by the wide community of applications developers.
Subroutine library
problem-solving library
C++ interval standard
1-0
Regular Paper
George F.
Corliss
George F. Corliss
R. Baker
Kearfott
R. Baker Kearfott
Ned
Nedialkov
Ned Nedialkov
John D.
Pryce
John D. Pryce
Spencer
Smith
Spencer Smith
10.4230/DagSemProc.06021.7
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Robustness and Randomness
Robustness problems of computational geometry algorithms is a topic that has been subject to intensive research efforts from both computer science and mathematics communities. Robustness problems are caused by the lack of precision in computations involving floating-point instead of real numbers. This paper reviews methods dealing with robustness and inaccuracy problems. It discussed approaches based on exact arithmetic, interval arithmetic and probabilistic methods. The paper investigates the possibility to use randomness at certain levels of reasoning to make geometric constructions more robust.
Robustness
interval
randomness
inaccuracy
geometric computation
1-23
Regular Paper
Dominique
Michelucci
Dominique Michelucci
Jean Michel
Moreau
Jean Michel Moreau
Sebti
Foufou
Sebti Foufou
10.4230/DagSemProc.06021.8
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Transfinite interpolation for well-definition in error analysis in solid modelling
An overall approach to the problem of error analysis in the context of solid modelling, analogous to the standard forward/backward error analysis of Numerical Analysis, was described in a recent paper by Hoffmann and Stewart. An important subproblem within this overall approach is the well-definition of the sets specified by inconsistent data. These inconsistencies may come from the use of finite-precision real-number arithmetic, from the use of low-degree curves to approximate boundaries, or from terminating an infinite convergent (subdivision) process after only a finite number of steps.
An earlier paper, by Andersson and the present authors, showed how to resolve this problem of well-definition, in the context of standard trimmed-NURBS representations, by using the Whitney Extension Theorem. In this paper we will show how an analogous approach can be used in the context of trimmed surfaces based on combined-subdivision representations, such as those proposed by Litke, Levin and SchrÃƒÂ¶der.
A further component of the problem of well-definition is ensuring that adjacent patches in a representation do not have extraneous intersections. (Here, "extraneous intersections" refers to intersections, between two patches forming part of the boundary, other than prescribed intersections along a common edge or at a common vertex.) The paper also describes the derivation of a bound for normal vectors that can be used for this purpose. This bound is relevant both in the case of trimmed-NURBS representations, and in the case of combined subdivision with trimming.
Forward/backward error analysis
robustness
well-definition
trimmed NURBS
combined subdivision
trimming
bounds on normals
1-12
Regular Paper
Neil
Stewart
Neil Stewart
Malika
Zidani
Malika Zidani
10.4230/DagSemProc.06021.9
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Upper and Lower Bounds on Sizes of Finite Bisimulations of Pfaffian Dynamical Systems
In this paper we study a class of dynamical systems defined by Pfaffian maps. It is a sub-class of o-minimal dynamical systems which capture rich
continuous dynamics and yet can be studied using finite bisimulations.
The existence of finite bisimulations for o-minimal dynamical and hybrid systems has been shown by several authors; see e.g. Brihaye et al (2004), Davoren (1999), Lafferriere et al (2000).
The next natural question to investigate is how the sizes of such bisimulations can be bounded. The first step in this direction was done by Korovina et al (2004) where a double exponential upper bound was shown for Pfaffian dynamical and hybrid systems. In the present paper we improve this bound to a single exponential upper bound. Moreover we show that this bound is tight in general, by exhibiting a parameterized class of systems on which the exponential bound is attained.
The bounds provide a basis for designing efficient algorithms for computing
bisimulations, solving reachability and motion planning problems.
Hybrid systems
Pfaffian functions
bisimulation
1-18
Regular Paper
Margarita
Korovina
Margarita Korovina
Nicolai
Vorobjov
Nicolai Vorobjov
10.4230/DagSemProc.06021.10
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Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format
We searched for the worst cases for correct rounding of the exponential
function in the IEEE 754r decimal64 format, and computed all the bad cases
whose distance from a breakpoint (for all rounding modes) is less than
$10^{-15}$,ulp, and we give the worst ones. In particular, the worst case
for $|x| geq 3 imes 10^{-11}$ is $exp(9.407822313572878 imes 10^{-2})
= 1.098645682066338,5,0000000000000000,278ldots$. This work can be
extended to other elementary functions in the decimal64 format and allows
the design of reasonably fast routines that will evaluate these functions
with correct rounding, at least in some domains.
Floating-point arithmetic
decimal arithmetic
table maker's dilemma
correct rounding
elementary functions
1-10
Regular Paper
Vincent
Lefèvre
Vincent Lefèvre
Damien
Stehlé
Damien Stehlé
Paul
Zimmermann
Paul Zimmermann
10.4230/DagSemProc.06021.11
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