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18
Interval Computations: Introduction, Uses, and Resources
 Euromath Bulletin
, 1996
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Arbitrary Precision Real Arithmetic: Design and Algorithms
, 1996
"... this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms fo ..."
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this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms for rational operations and transcendental functions for this representation. Finally we describe briefly the prototype written in Caml. 2. Computable real numbers
Consistency Techniques in Ordinary Differential Equations
, 2000
"... This paper takes a fresh look at the application of interval analysis to ordinary differential equations and studies how consistency techniques can help address the accuracy problems typically exhibited by these methods, while trying to preserve their efficiency. It proposes to generalize interval t ..."
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This paper takes a fresh look at the application of interval analysis to ordinary differential equations and studies how consistency techniques can help address the accuracy problems typically exhibited by these methods, while trying to preserve their efficiency. It proposes to generalize interval techniques intoatwostep process: a forward process that computes an enclosure and a backward process that reduces this enclosure. Consistency techniques apply naturally to the backward (pruning) step but can also be applied to the forward phase. The paper describes the framework, studies the various steps in detail, proposes a number of novel techniques, and gives some preliminary experimental results to indicate the potential of this new research avenue.
Guaranteed Error Bounds for Ordinary Differential Equations
 In Theory of Numerics in Ordinary and Partial Differential Equations
, 1994
"... Hamming once said, "The purpose of computing is insight, not numbers." If that is so, then the speed of our computers should be measured in insights per year, not operations per second. One key insight we wish from nearly all computing in engineering and scientific applications is, "H ..."
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Hamming once said, "The purpose of computing is insight, not numbers." If that is so, then the speed of our computers should be measured in insights per year, not operations per second. One key insight we wish from nearly all computing in engineering and scientific applications is, "How accurate is the answer?" Standard numerical analysis has developed techniques of forward and backward error analysis to help provide this insight, but even the best codes for computing approximate answers can be fooled. In contrast, validated computation ffl checks that the hypotheses of appropriate existence and uniqueness theorems are satisfied, ffl uses interval arithmetic with directed rounding to capture truncation and rounding errors in computation, and ffl organizes the computations to obtain as tight an enclosure of the answer as possible. These notes for a series of lectures at the VIth SERC Numerical Analysis Summer School, Leicester University, apply the principles of validated computatio...
A shooting approach to the Lorenz equations
 MR 93f:58150
"... Abstract. We announce and outline a proof of the existence of a homoclinic orbit of the Lorenz equations. In addition, we develop a shooting technique and two key conditions, which lead to the existence of a onetoone correspondence between a set of solutions and the set of all infinite sequences o ..."
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Abstract. We announce and outline a proof of the existence of a homoclinic orbit of the Lorenz equations. In addition, we develop a shooting technique and two key conditions, which lead to the existence of a onetoone correspondence between a set of solutions and the set of all infinite sequences of 1’s and 3’s. 1.
InC++ Library Family for Interval Computations
 INTERNATIONAL JOURNAL OF RELIABLE COMPUTING. SUPPLEMENT TO THE INTERNATIONAL WORKSHOP ON APPLICATIONS OF INTERVAL COMPUTATIONS
, 1995
"... This paper presents a series of C++ libraries for interval function evaluation and constraint satisfaction. Classical interval arithmetic (IA) (Moore, 1966) is extended by open ended intervals, the notion of infinity and by "complement" and discontinuous intervals. Both algebraic and numer ..."
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This paper presents a series of C++ libraries for interval function evaluation and constraint satisfaction. Classical interval arithmetic (IA) (Moore, 1966) is extended by open ended intervals, the notion of infinity and by "complement" and discontinuous intervals. Both algebraic and numerical IA techniques are combined for obtaining the actual range of interval functions efficiently and for determining better than local solutions for interval constraint satisfaction problems. Our practical goal is a set of portable C++ libraries that can be used in applications without deep understanding of interval analysis.
Interval Computations On The Spreadsheet
 Applications of Interval Computations
, 1996
"... This paper reviews work on using interval arithmetic as the basis for next generation spreadsheet programs capable of dealing with rounding errors, imprecise data, and numerical constraints. A series of ever more versatile computational models for spreadsheets are presented beginning from classical ..."
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This paper reviews work on using interval arithmetic as the basis for next generation spreadsheet programs capable of dealing with rounding errors, imprecise data, and numerical constraints. A series of ever more versatile computational models for spreadsheets are presented beginning from classical interval arithmetic and ending up with interval constraint satisfaction. In order to demonstrate the ideas, an actual implementation of each model as a class library is presented and its integration with a commercial spreadsheet program is explained. 1 LIMITATIONS OF SPREADSHEET COMPUTING Spreadsheet programs, such as MS Excel, Quattro Pro, Lotus 123, etc., are among the most widely used applications of computer science. Since the pioneering days of VisiCalc and others, spreadsheet programs have been enhanced immensely with new features. However, the underlying computational paradigm of evaluating arithmetical functions by using ordinary machine arithmetic has remained the same. The wor...
Multistep Filtering Operators for Ordinary Differential Equations
, 1999
"... Interval methods for ordinary differential equations (ODEs) provide guaranteed enclosures of the solutions and numerical proofs of existence and unicity of the solution. Unfortunately, they may result in large overapproximations of the solution because of the loss of precision in interval computat ..."
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Interval methods for ordinary differential equations (ODEs) provide guaranteed enclosures of the solutions and numerical proofs of existence and unicity of the solution. Unfortunately, they may result in large overapproximations of the solution because of the loss of precision in interval computations and the wrapping effect. The main open issue in this area is to find tighter enclosures of the solution, while not sacrificing efficiency too much. This paper takes a constraint satisfaction approach to this problem, whose basic idea is to iterate a forward step to produce an initial enclosure with a pruning step that tightens it. The paper focuses on the pruning step and proposes novel multistep ltering operators for ODEs. These operators are based on interval extensions of amultistep solution that are obtained by using (Lagrange and Hermite) interpolation polynomials and their error terms. The paper also shows how traditional techniques (such as meanvalue forms and coordinate transformations) can be adapted to this new context. Preliminary experimental results illustrate the potential of the approach, especially on stiff problems, wellknown to be very difficult to solve.
The constructive reals as a Java Library
 J. Log. Algebr. Program
, 2004
"... We describe an implementation of the computable (or constructive) real numbers as a pure Java library. To the user, the library interface appears very similar to that of some other numeric types provided by the standard Java library. The primary goal of the implementation is simplicity, so that the ..."
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We describe an implementation of the computable (or constructive) real numbers as a pure Java library. To the user, the library interface appears very similar to that of some other numeric types provided by the standard Java library. The primary goal of the implementation is simplicity, so that the implementation could be easily understood, and to allow simple informal correctness arguments. We hope to demonstrate that even such a basic implementation of constructive real arithmetic can be useful in a number of contexts, including in a desk calculator utility distributed with the package. A secondary goal was to demonstrate that some secondorder functions on the reals, such as restricted inverse and derivative operations, can be implemented with su#cient performance to be useful.
A Deterministic Descartes Algorithm for Real Polynomials
, 2008
"... We describe a Descartes algorithm for root isolation of polynomials with real coefficients. It is assumed that the coefficients of the polynomial can be approximated with arbitrary precision; exact computation in the field of coefficients is not required. We refer to such coefficients as bitstream c ..."
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We describe a Descartes algorithm for root isolation of polynomials with real coefficients. It is assumed that the coefficients of the polynomial can be approximated with arbitrary precision; exact computation in the field of coefficients is not required. We refer to such coefficients as bitstream coefficients. The algorithm is deterministic and has almost the same asymptotic complexity as the randomized bitstreamDescartes algorithm of Eigenwillig et al. (2005). Besides being deterministic, the algorithm is also somewhat simpler to analyze.