FILIB++, a fast interval library supporting containment computations

Lerch, Michael L. F; Tischler, German; Gudenberg, Jürgen Wolff von; Hofschuster, Werner; Krämer, Walter

doi:10.1145/1141885.1141893

Cited by 57 publications

(37 citation statements)

References 8 publications

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“…We emphasize that suitable Fourier and Lindstedt (Fourier-Taylor) models are ubiquitous in computer-assisted proofs in KAM theory and renormalization theory [13,19,18,42,43,49]. Our validation algorithms for proving the existence of invariant tori were implemented using our own C++ library to manipulate Fourier models together with the rigorous interval library FILIB++ Interval Library; see [48]. All the validations presented here were tested with several types of computers working under several operating systems, although we report only the results obtained with a machine with Intel Core2 Quad CPU Q9550 at 2.83 GHz working under Debian, using one of the processors.…”

Section: The Methodologymentioning

confidence: 99%

Reliable Computation of Robust Response Tori on the Verge of Breakdown

Figueras¹,

Haro²

2012

SIAM J. Appl. Dyn. Syst.

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Abstract. We prove the existence and local uniqueness of invariant tori on the verge of breakdown for two systems: the quasi-periodically driven logistic map and the quasi-periodically forced standard map. These systems exemplify two scenarios: the Heagy-Hammel route for the creation of strange nonchaotic attractors and the nonsmooth bifurcation of saddle invariant tori. Our proofs are computerassisted and are based on a tailored version of the Newton-Kantorovich theorem. The proofs cannot be performed using classical perturbation theory because the two scenarios are very far from the perturbative regime, and fundamental hypotheses such as reducibility or hyperbolicity either do not hold or are very close to failing. Our proofs are based on a reliable computation of the invariant tori and a careful study of their dynamical properties, leading to the rigorous validation of the numerical results with our novel computational techniques. Key words. normally hyperbolic invariant manifolds, computer validations, invariant tori, strange nonchaotic attractors AMS subject classifications. 37C55, 37D10, 65P99DOI. 10.1137/1008092221. Introduction. The goal of this paper is to present a new methodology to provide rigorous proofs of the existence and local uniqueness of (fiberwise hyperbolic) invariant tori in quasi-periodic systems, even in cases in which the available perturbative theory does not apply. The methodology is suitable for computer-assisted proofs and consists in checking the hypotheses of a validation result based on the Newton-Kantorovich theorem [27]. As an application of the methodology, we prove the existence and local uniqueness of invariant tori on the verge of breakdown in two scenarios: the Heagy-Hammel route to strange nonchaotic attractors (SNA) [32] in a quasi-periodically driven logistic map and the breakdown of saddle tori [26] in a quasi-periodically forced standard map.

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Section: The Methodologymentioning

confidence: 99%

Reliable Computation of Robust Response Tori on the Verge of Breakdown

Figueras¹,

Haro²

2012

SIAM J. Appl. Dyn. Syst.

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“…The library filib++ is known as an efficient portable C++ library supporting interval arithmetic and fast computation of guaranteed bounds for interval versions of a comprehensive set of elementary functions [9], [10]. There are two specific features of the library regarding the representation of real intervals.…”

Section: Mathlink Connection To Filib++mentioning

confidence: 99%

“…possess several features not attributable to the compiled languages and on the other hand most of the interval software is developed in some compiled language for efficiency reasons, it is interesting to study the possibilities for interoperability between these two kinds of interval supporting environments. In this paper we focus on the interaction between Mathematica [17] and two external C++ libraries for interval computations, filib++ [9], [10] and C-XSC [4], [7] via MathLink communication protocol [5], [15]. In particular, we present how to call external programs based on either of these two interval libraries from within a Mathematica session, thus exchanging data without using intermediate files but under dynamics and interactivity in the communication.…”

Section: Introductionmentioning

confidence: 99%

Mathematica Connectivity to Interval Libraries filib++ and C-XSC

Popova

2009

Numerical Validation in Current Hardware Architectures

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Abstract. Building interval software interoperability can be a good solution when re-using high-quality legacy code or when accessing functionalities unavailable natively in one of the software packages. In this work we present the integration of programs based on the interval libraries filib++ and C-XSC into Mathematica via MathLink communication protocol. On some small easily readable programs we demonstrate: i) some details of MathLink technology, ii) the transparency of numerical data communication without any conversion, iii) the advantage of symbolic manipulation interfaces -the access to the external compiled language functionality from within Mathematica is often even more convenient than from its own native environment.

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“…In the containment set approach [9,8], how it is implemented in filib++ [3], e.g., infinities are treated as values. There are point intervals and operations for ±∞.…”

Section: The Containment Set Approachmentioning

confidence: 99%

Numerical Validation in Current Hardware Architectures

Cuyt¹,

Krämer²,

Luther³

et al. 2009

Lecture Notes in Computer Science

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Interval arithmetic is arithmetic for continuous sets. Floating-point intervals are intervals of real numbers with floating-point bounds. Operations for intervals can be efficiently implemented. Hence, the time is ripe for standardization. In this paper we present an interval model that is mathematically sound and closed for the 4 basic operations. The model allows for exception free interval arithmetic, if we carefully distinguish between clean and reliable interval arithmetic on one side and rounded floating-point arithmetic on the other side. Elementary functions for intervals can be defined. In some application areas loose evaluation of functions, i.e. evaluation over an interval which is not completely contained in the function domain, is recommended, In this case, however, a discontinuity flag has to be set to inform that Brouwer's fixed point theorem is no longer applicable. Real Interval Arithmetic Real Interval ArithmeticReal interval arithmetic is defined as arithmetic on continuous (in the sense of complete, not discrete) sets. Remark 2 If f is continuous, the range is an interval.

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FILIB++, a fast interval library supporting containment computations

Cited by 57 publications

References 8 publications

Reliable Computation of Robust Response Tori on the Verge of Breakdown

Reliable Computation of Robust Response Tori on the Verge of Breakdown

Mathematica Connectivity to Interval Libraries filib++ and C-XSC

Numerical Validation in Current Hardware Architectures

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