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Guaranteed Proofs Using Interval Arithmetic

M. Daumas, G. Melquiond, C. Munoz
17th IEEE Symposium on Computer Arithmetic (ARITH'05)  
This paper presents a set of tools for mechanical reasoning of numerical bounds using interval arithmetic.  ...  This paper presents a set of tools for mechanical reasoning of numerical bounds using interval arithmetic.  ...  Acknowledgment Proofs of Sections 4 and 5 were checked on high performance clusters set up and maintained by the Reso research project of the LIP computer science laboratory.  ... 
doi:10.1109/arith.2005.25 dblp:conf/arith/DaumasMM05 fatcat:obkunjnxhfgo7ihyuspmoluuvq

Enclosing Solutions of Linear Equations

Jiri Rohn, Georg Rex
1998 SIAM Journal on Numerical Analysis  
using Brouwer's fixed-point theorem.  ...  It is shown that Rump's method for enclosing solutions of linear equations can be reformulated in an interval-free form and that the underlying inclusion result can be proved by elementary means without  ...  When using the interval arithmetic operations, downwardly and upwardly oriented rounding must be used to guarantee that Y ⊂ Int(X) holds; then Y is a verified enclosure of the solution x * .  ... 
doi:10.1137/s0036142996299423 fatcat:vvvvwohw7jap7nhnjabdkekvxi

Verified Real Number Calculations: A Library for Interval Arithmetic [article]

Marc Daumas , César Muñoz
2007 arXiv   pre-print
In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and taylor series expansions.  ...  Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theorem-prover interaction.  ...  It guarantees that evaluations of an expression using interval arithmetic bound its exact real value.  ... 
arXiv:0708.3721v1 fatcat:5a6672hxfzhlrnavvchbzlc7aq

Verified Real Number Calculations: A Library for Interval Arithmetic

M. Daumas, D. Lester, C. Muoz
2009 IEEE transactions on computers  
In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and taylor series expansions.  ...  Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theorem-prover interaction.  ...  It guarantees that evaluations of an expression using interval arithmetic bound its exact real value.  ... 
doi:10.1109/tc.2008.213 fatcat:4aadux52sncpjan3yjo7fkauja

Real Number Calculations and Theorem Proving [chapter]

David R Lester
2008 Lecture Notes in Computer Science  
Then, based on these bounds, we develop a rational interval arithmetic where real number calculations can be performed in an algebraic setting.  ...  Wouldn't it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework.  ...  Acknowledgment The authors would like to thank Marc Daumas for his early interest on this work, and his key suggestion to use interval arithmetic to automate our initial lower/upper bound technique.  ... 
doi:10.1007/978-3-540-71067-7_19 fatcat:lccfqatnmrg5fal3zvorrlktuu

Real Number Calculations and Theorem Proving [chapter]

César Muñoz, David Lester
2005 Lecture Notes in Computer Science  
Then, based on these bounds, we develop a rational interval arithmetic where real number calculations can be performed in an algebraic setting.  ...  Wouldn't it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework.  ...  Acknowledgment The authors would like to thank Marc Daumas for his early interest on this work, and his key suggestion to use interval arithmetic to automate our initial lower/upper bound technique.  ... 
doi:10.1007/11541868_13 fatcat:isfmzar76bh3rhuq6c6npolepm

Interval methods that are guaranteed to underestimate (and the resulting new justification of Kaucher arithmetic)

Vladik Kreinovich, Vyacheslav M. Nesterov, Nina A. Zheludeva
1996 Reliable Computing  
Proposition 1 gives a mathematical proof of these estimates being the best; this proof is not so easy as the proofs of many algebraic results about interval arithmetic because we have to consider all possible  ...  with traditional interval arithmetic; operations between improper intervals; these operations are equivalent to operations of traditional interval arithmetic; operations between proper and improper intervals  ... 
doi:10.1007/bf02425913 fatcat:qqcwxasvvjhbrhynmb3b4ne5ti

Interval methods for verifying structural optimality of circle packing configurations in the unit square

Mihály Csaba Markót
2007 Journal of Computational and Applied Mathematics  
The required statements are verified with mathematical rigor using interval arithmetic tools.  ...  Recently, a global optimization method based exclusively on interval arithmetic calculations has been designed for this problem.  ...  In our studies the necessary numerical computations are performed using interval arithmetic, which enables us to compute reliable bounds on the results of floating point calculations, and thus to prove  ... 
doi:10.1016/j.cam.2005.08.039 fatcat:ycxy53vjpjeuhowtcxr6ykkf4y

Guaranteed state estimation by zonotopes

T. Alamo, J.M. Bravo, E.F. Camacho
2005 Automatica  
Interval arithmetic is used to calculate a guaranteed trajectory of the state process. Two examples have been provided for clarifying the algorithm.  ...  This paper presents a new approach to guaranteed state estimation for non linear discrete-time systems with a bounded description of noise and parameters.  ...  Each sample time, a guaranteed bound of the uncertain trajectory of the system is calculated using interval arithmetic.  ... 
doi:10.1016/j.automatica.2004.12.008 fatcat:wppuffwcg5hqxdst2v4gokzqry

An overview of the upcoming IEEE P-1788 working group document: Standard for interval arithmetic

Ralph Baker Kearfott
2013 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS)  
We provide a summary of the goals, underlying philosophy, work, decisions, product, and completion schedule of the IEEE P-1788 working group on interval arithmetic.  ...  This leads to the ability to use floating point computations to provide mathematical proofs of appropriately posed assertions, to provide guarantees that a system subject to manufacturing uncertainties  ...  Such proofs are largely effected by computing mathematically rigorous upper and lower bounds on the range of a function over hyper-rectangles, using the fundamental theorem of interval arithmetic: Theorem  ... 
doi:10.1109/ifsa-nafips.2013.6608444 dblp:conf/ifsa/Kearfott13 fatcat:f4govcc2qzb3rj4kwb3zzj4she

Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

Érik Martin-Dorel, Guillaume Melquiond
2015 Journal of automated reasoning  
It is based on a formalization of floating-point and interval arithmetic, associated with an on-the-fly computation of Taylor expansions.  ...  In fact, the inherent difficulty of computing such bounds often mandates a formal proof of them.  ...  Computations are performed using interval arithmetic. For multivariate polynomial expressions, the PVS proof system uses an approach based on Bernstein polynomials [23] .  ... 
doi:10.1007/s10817-015-9350-4 fatcat:4ges4bruovhqbphmqstyfhqkz4

Interval computation as deduction in chip

J.H.M. Lee, M.H. Van Emden
1993 The Journal of Logic Programming  
To reduce arithmetic primitives with interval domains, we use our interval narrowing technique as an implementation of the looking-ahead inference rule.  ...  Cleary proposed a relational form of interval arithmetic that was incorporated in BNR Prolog in such a way that variables already bound can be bound again.  ...  Interval arithmetic contributes methods guaranteeing correctness at the level of a simple arithmetic expression.  ... 
doi:10.1016/0743-1066(93)90045-i fatcat:wva2qfziurfsfddc2fglybk2zm

An intervalal gorithm for solving systems of linear equations to prespecified accuracy

J. W. Demmel, F. Krückeberg
1985 Computing  
We describe an interval arithmetic algorithm for solving a special class of simultaneous linear equations.  ...  The algorithm uses fixed point arithmetic, and has two properties which distinguish it from earlier algorithms: given the absolute accuracy a desired, the algorithm uses only as much precision as needed  ...  Kriickeberg's students who implemented the fixed point arithmetic and assisted in programming the algorithm.  ... 
doi:10.1007/bf02259840 fatcat:5rzcjs7vg5clvciywd354ow26m

Nonstandard methods and the Erdős-Turán conjecture [chapter]

Steven C. Leth
2007 The Strength of Nonstandard Analysis  
Here we recall that a < t, d, w > cell is merely a collection of t intervals in arithmetic progression on which A is nonempty. Proof.  ...  A complete proof of this theorem appears in [8] , but we will outline the proof here, as it provides the clearest illustration of how the use of the nonstandard model provides us with a new set of tools  ... 
doi:10.1007/978-3-211-49905-4_9 fatcat:unkvmc3xcfas3e2fqekq2pv6ye

Large Subsets of Euclidean Space Avoiding Infinite Arithmetic Progressions [article]

Laurestine Bradford, Hannah Kohut, Yuveshen Mooroogen
2022 arXiv   pre-print
We prove that this result does not extend to infinite arithmetic progressions in the following sense: for each λ in [0,1), we construct a subset of ℝ that intersects every interval of unit length in a  ...  set of measure at least λ, but that does not contain any infinite arithmetic progression.  ...  The set does not contain any arithmetic progression with integer gap size. Proof.  ... 
arXiv:2205.04786v1 fatcat:64ihdtgs3nghxngoixsyu7n4ga
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