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16
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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Cited by 19 (0 self)
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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
An overview of semantics for the validation of numerical programs
, 2005
"... Abstract. In this article, we introduce a simple formal semantics for floatingpoint numbers with errors which is expressive enough to be formally compared to the other methods. Next, we define formal semantics for interval, stochastic, automatic differentiation and error series methods. This enable ..."
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Cited by 15 (5 self)
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Abstract. In this article, we introduce a simple formal semantics for floatingpoint numbers with errors which is expressive enough to be formally compared to the other methods. Next, we define formal semantics for interval, stochastic, automatic differentiation and error series methods. This enables us to formally compare the properties calculated in each semantics to our reference, simple semantics. Most of these methods having been developed to verify numerical intensive codes, we also discuss their adequacy to the formal validation of softwares and to static analysis. Finally, this study is completed by experimental results. 1
Semantics of roundoff error propagation in finite precision computations
 Journal of Higher Order and Symbolic Computation
, 2006
"... Abstract. We introduce a concrete semantics for floatingpoint operations which describes the propagation of roundoff errors throughout a calculation. This semantics is used to assert the correctness of a static analysis which can be straightforwardly derived from it. In our model, every elementary ..."
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Cited by 11 (6 self)
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Abstract. We introduce a concrete semantics for floatingpoint operations which describes the propagation of roundoff errors throughout a calculation. This semantics is used to assert the correctness of a static analysis which can be straightforwardly derived from it. In our model, every elementary operation introduces a new first order error term, which is later propagated and combined with other error terms, yielding higher order error terms. The semantics is parameterized by the maximal order of error to be examined and verifies whether higher order errors actually are negligible. We consider also coarser semantics computing the contribution, to the final error, of the errors due to some intermediate computations. As a result, we obtain a family of semantics and we show that the less precise ones are abstractions of the more precise ones.
Static Analyses of FloatingPoint Operations
 In SAS’01, volume 2126 of LNCS
, 2001
"... Computers manipulate approximations of real numbers, called floatingpoint numbers. The calculations they make are accurate enough for most applications. Unfortunately, in some (catastrophic) situations, the floatingpoint operations lose so much precision that they quickly become irrelevant. In thi ..."
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Cited by 8 (0 self)
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Computers manipulate approximations of real numbers, called floatingpoint numbers. The calculations they make are accurate enough for most applications. Unfortunately, in some (catastrophic) situations, the floatingpoint operations lose so much precision that they quickly become irrelevant. In this article, we review some of the problems one can encounter, focussing on the IEEE7541985 norm. We give a (sketch of a) semantics of its basic operations then abstract them (in the sense of abstract interpretation) to extract information about the possible loss of precision. The expected application is abstract debugging of software ranging from simple onboard systems (which use more and more ontheshelf microprocessors with floatingpoint units) to scientific codes. The abstract analysis is demonstrated on simple examples and compared with related work. 1
Dynamical control of computations using the trapezoidal and Simpson's rules
 Journal of Universal Computer Science
, 1998
"... Abstract: If In is the approximation of a de nite integral I = Z b a f(x)dx with step b, a 2n using the trapezoidal rule (respectively Simpson's rule), if Ca;b denotes the number of signi cant digits common to a and b, we show, in this paper, that CIn;In+1 = 4 1 ..."
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Cited by 5 (1 self)
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Abstract: If In is the approximation of a de nite integral I = Z b a f(x)dx with step b, a 2n using the trapezoidal rule (respectively Simpson's rule), if Ca;b denotes the number of signi cant digits common to a and b, we show, in this paper, that CIn;In+1 = 4 1
Dynamic restarting strategy for a BICGSTAB algorithm using discrete stochastic arithmetic
"... Introduction A lot of scientic elds require a resolution of a linear equation system. It gives often rise to a very large sparse unsymmetric matrix. Sometimes, we can only access its elements implicitly that is to say we only know how to make operations on it as matrixvector product. In these case ..."
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Introduction A lot of scientic elds require a resolution of a linear equation system. It gives often rise to a very large sparse unsymmetric matrix. Sometimes, we can only access its elements implicitly that is to say we only know how to make operations on it as matrixvector product. In these cases, direct methods (e.g., gauss) can not be applied and iterative lanczostype methods [6, 5] are preferred to classical stationary ones (e.g., sor) that require too restrictive matrix properties. Unfortunately, they are seriously prone to roundo errors for large and badly scaled matrix and the convergence tends to be very erratic. Is the latter phenomenon a mathematically foreseeable event and in which measure can we rely on the oatingpoint arithmetic? In order to control the computation course, we have used the discrete stochastic arithmetic [3]. Based on a probabilistic approach of rounding errors, it allows us to estimate the p
The influence of system calls and interrupts on the performance of a PC cluster using a remote DMA communication primitive
, 2002
"... This paper presents an efficient MPI implementation on a cluster of PCs using a remote DMA communication primitive. For experimental purposes, the MultiPC (MPC) parallel computer was used. It consists of standard PCs interconnected through a gigabit High Speed Link (HSL) network. This paper focuses ..."
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This paper presents an efficient MPI implementation on a cluster of PCs using a remote DMA communication primitive. For experimental purposes, the MultiPC (MPC) parallel computer was used. It consists of standard PCs interconnected through a gigabit High Speed Link (HSL) network. This paper focuses on communication software layers over the HSL network. Two implementations of MPI are described. The first one uses hardware interrupts for network events signaling and system calls in the communication critical path. The second one is based on full userlevel communications. Measures show a latency of 15 s on a Pentium II350 with this optimized implementation. A quantitative analysis shows how system calls and interrupts impact on communication time. To tally performance in a realistic environment, experiments were run on the Gauss elimination method using a parallel implementation of a local numerical analysis computational package (CADNA).
In Pursuit of Real Answers ∗
"... Digital computers permeate our physical world. This phenomenon creates a pressing need for tools that help us understand a priori how digital computers can affect their physical environment. In principle, simulation can be a powerful tool for animating models of the world. Today, however, there is n ..."
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Digital computers permeate our physical world. This phenomenon creates a pressing need for tools that help us understand a priori how digital computers can affect their physical environment. In principle, simulation can be a powerful tool for animating models of the world. Today, however, there is not a single simulation environment that comes with a guarantee that the results of the simulation are determined purely by a realvalued model and not by artifacts of the digitized implementation. As such, simulation with guaranteed fidelity does not yet exist. Towards addressing this problem, we offer an expository account of what is known about exact real arithmetic. We argue that this technology, which has roots that are over 200 years old, bears significant promise as offering exactly the right technology to build simulation environments with guaranteed fidelity. And while it has only been sparsely studied in this large span of time, there are reasons to believe that the time is right to accelerate research in this direction. ∗ This research was sponsored by the NSF under Award 0439017,
An introduction to the quality of computed solutions
 ACCURACY AND RELIABILITY IN SCIENTIFIC COMPUTING
, 2005
"... ..."
The CELL parallel processor has been jointly designed by IBM, Sony a...
"... Several super computers have been designed as massively parallel computers using the CELL processor as their main component. Such is for example the IBM Roadrunner which broke the world computing speed record in June 2008. However, even if machines of this kind are absolutely necessary to solve nume ..."
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Several super computers have been designed as massively parallel computers using the CELL processor as their main component. Such is for example the IBM Roadrunner which broke the world computing speed record in June 2008. However, even if machines of this kind are absolutely necessary to solve numerical problems that could not be solved otherwise, the question of the accuracy of the solution may become critical when obtained with a monstrous amount of computation. Concerning the question of accuracy, the arithmetic of the eight on chip parallel processors of CELL have two drawbacks: i) rounding is towards zero and not to nearest, ii) division is very inaccurate. The paper deals with the effect of these two particularities on the result of scientific computations. First, it is shown that the classical computation of the inner product of two ndimensional vectors has an accuracy which is O ( √ n) for rounding to nearest and O(n) for all other rounding modes. Thus the fast rounding to