60 research outputs found
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-preci- sion interval arithmetic, on a specific problem, is presented: it shows that the overhead due to the multiple precision is completely admissible. 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.Cet article justifie le besoin d’une arithmétique par intervalles en précision arbitraire : pour fournir des résultats précis, un calcul par intervalles requiert des intervalles en entrée qui soient fins ; c’est pour cette raison que la bissection est un procédé si souvent employé dans les algorithmes par intervalles. La bibliothèque MPFI a été développée pour répondre à ce besoin : en effet, aucune bibliothèque existante n’offrait de spécifications satisfaisantes. Les caractéristiques de cette bibliothèque sont rapidement données puis une comparaison avec une bibliothèque d’arithmétique par intervalles en précision fixée est menée sur un problème spécifique : elle met en évidence le fait que le surcoût lié à la gestion de la précision multiple est tout à fait acceptable. Pour terminer, quelques applications basées sur MPFI sont présentées : robotique, isolation des racines réelles de polynômes (par un algorithme combinant calcul symbolique et calcul numérique) et approximation avec une précision arbitraire de zéros réel
Implementing an interval computation library for OCaml on x86/amd64 architectures
International audienceIn this paper, we present two implementations of interval arithmetic for OCaml on x86/amd64 architectures. The first implementation is a binding to the classical MPFI/MPFR library. It provides access to multi-precision floating-point arithmetic and multi-precision floating-point intervalarithmetic. The second implementation has been natively written in assembly language for low-level functions and in OCaml for higher-levelfunctions. It has proven as fast as classical C or C++ implementations
Computational Complexity of Iterated Maps on the Interval (Extended Abstract)
The exact computation of orbits of discrete dynamical systems on the interval
is considered. Therefore, a multiple-precision floating point approach based on
error analysis is chosen and a general algorithm is presented. The correctness
of the algorithm is shown and the computational complexity is analyzed. As a
main result, the computational complexity measure considered here is related to
the Ljapunow exponent of the dynamical system under consideration
A Subdivision Solver for Systems of Large Dense Polynomials
We describe here the package {\tt subdivision\\_solver} for the mathematical
software {\tt SageMath}. It provides a solver on real numbers for square
systems of large dense polynomials. By large polynomials we mean multivariate
polynomials with large degrees, which coefficients have large bit-size. While
staying robust, symbolic approaches to solve systems of polynomials see their
performances dramatically affected by high degree and bit-size of input
polynomials.Available numeric approaches suffer from the cost of the evaluation
of large polynomials and their derivatives.Our solver is based on interval
analysis and bisections of an initial compact domain of where solutions
are sought. Evaluations on intervals with Horner scheme is performed by the
package {\tt fast\\_polynomial} for {\tt SageMath}.The non-existence of a
solution within a box is certified by an evaluation scheme that uses a Taylor
expansion at order 2, and existence and uniqueness of a solution within a box
is certified with krawczyk operator.The precision of the working arithmetic is
adapted on the fly during the subdivision process and we present a new
heuristic criterion to decide if the arithmetic precision has to be increased
About the ''accurate mode'' of the IEEE 1788-2015 standard for interval arithmetic
The IEEE 1788-2015 standard for interval arithmetic defines three accuracy
modes for the so-called set-based flavor: tightest, accurate and valid. This
work in progress focuses on the accurate mode.First, an introduction to
interval arithmetic and to the IEEE 1788-2015 standard is given, then the
accurate mode is defined. How can this accurate mode be tested, when a library
implementing interval arithmetic claims to provide this mode? The chosen
approach is unit testing, and the elaboration of testing pairs for this
approach is developed.A discussion closes this paper: how can the tester be
tested? And if we go to the roots of the subject, is the accurate mode really
relevant or should it be dropped off in the next version of the standard
A framework to test interval arithmetic libraries and their IEEE 1788-2015 compliance
As developers of libraries implementing interval arithmetic, we faced the
same difficulties when it comes to testing our libraries. What must be tested?
How can we devise relevant test cases for unit testing? How can we ensure a
high (and possibly 100%) test coverage? Before considering these questions, we
briefly recall the main features of interval arithmetic and of the IEEE
1788-2015 standard for interval arithmetic. After listing the different aspects
that, in our opinion, must be tested, we contribute a first step towards
offering a test suite for an interval arithmetic library. First we define a
format that enables the exchange of test cases, so that they can be read and
tried easily. Then we offer a first set of test cases, for a selected set of
mathematical functions. Next, we examine how the Julia interval arithmetic
library, IntervalArithmetic.jl, actually performs to these tests. As this is an
ongoing work, we list extra tests that we deem important to perform.Comment: 2 figure
Computing Real Roots of Real Polynomials ... and now For Real!
Very recent work introduces an asymptotically fast subdivision algorithm,
denoted ANewDsc, for isolating the real roots of a univariate real polynomial.
The method combines Descartes' Rule of Signs to test intervals for the
existence of roots, Newton iteration to speed up convergence against clusters
of roots, and approximate computation to decrease the required precision. It
achieves record bounds on the worst-case complexity for the considered problem,
matching the complexity of Pan's method for computing all complex roots and
improving upon the complexity of other subdivision methods by several
magnitudes.
In the article at hand, we report on an implementation of ANewDsc on top of
the RS root isolator. RS is a highly efficient realization of the classical
Descartes method and currently serves as the default real root solver in Maple.
We describe crucial design changes within ANewDsc and RS that led to a
high-performance implementation without harming the theoretical complexity of
the underlying algorithm.
With an excerpt of our extensive collection of benchmarks, available online
at http://anewdsc.mpi-inf.mpg.de/, we illustrate that the theoretical gain in
performance of ANewDsc over other subdivision methods also transfers into
practice. These experiments also show that our new implementation outperforms
both RS and mature competitors by magnitudes for notoriously hard instances
with clustered roots. For all other instances, we avoid almost any overhead by
integrating additional optimizations and heuristics.Comment: Accepted for presentation at the 41st International Symposium on
Symbolic and Algebraic Computation (ISSAC), July 19--22, 2016, Waterloo,
Ontario, Canad
The MPFI Library: Towards IEEE 1788-2015 Compliance
International audienceThe IEEE 1788-2015 has standardized interval arithmetic. However, few libraries for interval arithmetic are compliant with this standard. The main features of the IEEE 1788-2015 standard are detailed, namely the structure into 4 levels, the possibility to accomodate a new mathematical theory of interval arithmetic through the notion of flavor, and the mechanism of decoration for handling exceptions. These features were not present in the libraries developed prior to the elaboration of the standard. MPFI is such a library: it is a C library, based on MPFR, for arbitrary precision interval arithmetic. MPFI is not (yet) compliant with the IEEE 1788-2015 standard for interval arithmetic: the planned modifications are presented. Some considerations about performance and HPC on interval computations based on this standard, or on MPFI, conclude the paper
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