11,313 research outputs found
Lazy algorithms for exact real arithmetic
In this article we propose a new representation for the real numbers. This representation can be conveniently used to implement exact real number computation with a lazy programming languages. In fact the new representation permits the exploitation of hardware implementation of arithmetic functions without generating the granularity problem. Moreover we present a variation of the Karatsuba algorithm for multiplication of integers. The new algorithm performs exact real number multiplication in a lazy way and has a lower complexity than the standard algorithm. \ua9 2004 Elsevier B.V. All rights reserved
Formal verification of exact computations using Newton's method
International audienceWe are interested in the certification of Newton's method. We use a formalization of the convergence and stability of the method done with the axiomatic real numbers of Coq's Standard Library in order to validate the computation with Newton's method done with a library of exact real arithmetic based on co-inductive streams. The contribution of this work is twofold. Firstly, based on Newton's method, we design and prove correct an algorithm on streams for computing the root of a real function in a lazy manner. Secondly, we prove that rounding at each step in Newton's method still yields a convergent process with an accurate correlation between the precision of the input and that of the result. An algorithm including rounding turns out to be much more efficient
Certified lattice reduction
Quadratic form reduction and lattice reduction are fundamental tools in
computational number theory and in computer science, especially in
cryptography. The celebrated Lenstra-Lenstra-Lov\'asz reduction algorithm
(so-called LLL) has been improved in many ways through the past decades and
remains one of the central methods used for reducing integral lattice basis. In
particular, its floating-point variants-where the rational arithmetic required
by Gram-Schmidt orthogonalization is replaced by floating-point arithmetic-are
now the fastest known. However, the systematic study of the reduction theory of
real quadratic forms or, more generally, of real lattices is not widely
represented in the literature. When the problem arises, the lattice is usually
replaced by an integral approximation of (a multiple of) the original lattice,
which is then reduced. While practically useful and proven in some special
cases, this method doesn't offer any guarantee of success in general. In this
work, we present an adaptive-precision version of a generalized LLL algorithm
that covers this case in all generality. In particular, we replace
floating-point arithmetic by Interval Arithmetic to certify the behavior of the
algorithm. We conclude by giving a typical application of the result in
algebraic number theory for the reduction of ideal lattices in number fields.Comment: 23 page
From coinductive proofs to exact real arithmetic: theory and applications
Based on a new coinductive characterization of continuous functions we
extract certified programs for exact real number computation from constructive
proofs. The extracted programs construct and combine exact real number
algorithms with respect to the binary signed digit representation of real
numbers. The data type corresponding to the coinductive definition of
continuous functions consists of finitely branching non-wellfounded trees
describing when the algorithm writes and reads digits. We discuss several
examples including the extraction of programs for polynomials up to degree two
and the definite integral of continuous maps
Engineering Art Galleries
The Art Gallery Problem is one of the most well-known problems in
Computational Geometry, with a rich history in the study of algorithms,
complexity, and variants. Recently there has been a surge in experimental work
on the problem. In this survey, we describe this work, show the chronology of
developments, and compare current algorithms, including two unpublished
versions, in an exhaustive experiment. Furthermore, we show what core
algorithmic ingredients have led to recent successes
Coinductive Formal Reasoning in Exact Real Arithmetic
In this article we present a method for formally proving the correctness of
the lazy algorithms for computing homographic and quadratic transformations --
of which field operations are special cases-- on a representation of real
numbers by coinductive streams. The algorithms work on coinductive stream of
M\"{o}bius maps and form the basis of the Edalat--Potts exact real arithmetic.
We use the machinery of the Coq proof assistant for the coinductive types to
present the formalisation. The formalised algorithms are only partially
productive, i.e., they do not output provably infinite streams for all possible
inputs. We show how to deal with this partiality in the presence of syntactic
restrictions posed by the constructive type theory of Coq. Furthermore we show
that the type theoretic techniques that we develop are compatible with the
semantics of the algorithms as continuous maps on real numbers. The resulting
Coq formalisation is available for public download.Comment: 40 page
Type classes for efficient exact real arithmetic in Coq
Floating point operations are fast, but require continuous effort on the part
of the user in order to ensure that the results are correct. This burden can be
shifted away from the user by providing a library of exact analysis in which
the computer handles the error estimates. Previously, we [Krebbers/Spitters
2011] provided a fast implementation of the exact real numbers in the Coq proof
assistant. Our implementation improved on an earlier implementation by O'Connor
by using type classes to describe an abstract specification of the underlying
dense set from which the real numbers are built. In particular, we used dyadic
rationals built from Coq's machine integers to obtain a 100 times speed up of
the basic operations already. This article is a substantially expanded version
of [Krebbers/Spitters 2011] in which the implementation is extended in the
various ways. First, we implement and verify the sine and cosine function.
Secondly, we create an additional implementation of the dense set based on
Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order
on undecidable structures, while it was limited to decidable structures before.
This hierarchy, based on type classes, allows us to share theory on the
naturals, integers, rationals, dyadics, and reals in a convenient way. Finally,
we obtain another dramatic speed-up by avoiding evaluation of termination
proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275
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