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A floating-point arithmetic ERROR ANALYSIS OF DIRECT AND INDIRECT COEFFICIENT UPDATING TECHNIQUES FOR ADAPTIVE LATTICE FILTERS

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 1993年第5期41卷 1809-1823页

作者： NORTH, RC ZEIDLER, JR KU, WH ALBERT, TR UNIV CALIF SAN DIEGO DEPT ELECT & COMP ENGN LA JOLLA CA 92039 USA UNIV CALIF SAN DIEGO DEPT ELECT & COMP ENGN LA JOLLA CA 92039 USA

This paper shows how finite precision arithmetic effects can deleteriously manifest themselves in both the stochastic gradient and the recursive least squares adaptive lattice filters. Closed form expressions are derived for the steady-state variance of the accumulated arithmetic error in a single adaptive lattice coefficient using a floating-point stochastic arithmetic error analysis. Emphasis is placed on the computational form or the time-recursive adaptive lattice coefficients. The analytical results show that the performance of adaptive lattice filters using a direct updating computational form is less sensitive to finite precision effects than that of adaptive lattice filters using an indirect updating computational form. This is shown to be the result of an accumulated arithmetic error that is reduced in the directly updated filter coefficients by the division of an unnormalized residual power estimate. In addition, a method for reducing the self-generated noise is presented. Experimental results obtained on a 32-b floating-point hardware implementation of the adaptive lattice filters and with computer simulations are included to verify the analytical results describing the effects of finite precision on adaptive lattice filters.

关键词： floating-point arithmetic Error analysis Lattices Adaptive filters Stochastic processes Least squares methods Steady-state Stochastic resonance Performance analysis Noise reduction

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floating-point arithmetic FOR COMPUTATIONAL GEOMETRY PROBLEMS WITH UNCERTAIN DATA

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INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS 2009年第4期19卷 371-385页

作者： Jiang, D. Stewart, N. F. Univ Montreal DIRO Montreal PQ Canada

It has been suggested in the literature that ordinary finite-precision floating-point arithmetic is inadequate for geometric computation, and that researchers in numerical analysis may believe that the difficulties of error in geometric computation can be overcome by simple approaches. It is the purpose of this paper to show that these suggestions, based on an example showing failure of a certain algorithm for computing planar convex hulls, are misleading, and why this is so. It is first shown how the now-classical backward error analysis can be applied in the area of computational geometry. This analysis is relevant in the context of uncertain data, which may well be the practical context for computational-geometry algorithms such as, say, those for computing convex hulls. The exposition will illustrate the fact that the backward error analysis does not pretend to over come the problem of finite precision: it merely provides a way to distinguish those algorithms that overcome the problem to whatever extent it is possible to do so. It is then shown that often the situation in computational geometry is exactly parallel to other areas, such as the numerical solution of linear equations, or the algebraic eigenvalue problem. Indeed, the example mentioned can be viewed simply as an example of the use of an unstable algorithm, for a problem for which computational geometry has already discovered provably stable algorithms. Finally, the paper discusses the implications of these analyses for applications in three-dimensional solid modeling. This is done by considering a problem defined in terms of a simple extension of the planar convex-hull algorithm, namely, the verification of the well-formedness of extruded objects. A brief discussion concerning more difficult problems in solid modeling is also included

关键词： floating-point arithmetic robustness in geometric computation stability planar convex hull backward error analysis

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floating-point arithmetic in the Coq system

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INFORMATION AND COMPUTATION 2012年 216卷 14-23页

作者： Melquiond, Guillaume Univ Paris 11 PCRI INRIA Saclay France F-91405 Orsay France

The process of proving some mathematical theorems can be greatly reduced by relying on numerically-intensive computations with a certified arithmetic. This article presents a formalization of floating-point arithmetic that makes it possible to efficiently compute inside the proofs of the Coq system. This certified library is a multi-radix and multi-precision implementation free from underflow and overflow. It provides the basic arithmetic operators and a few elementary functions. (C) 2012 Elsevier Inc. All rights reserved.

关键词： floating-point arithmetic Formal proofs Coq system

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Hierarchical search algorithm for error detection in floating-point arithmetic expressions

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JOURNAL OF SUPERCOMPUTING 2024年第1期80卷 1183-1205页

作者： Zhang, Zuoyan Xu, Jinchen Hao, Jiangwei Qu, Yang He, Haotian Zhou, Bei Informat Engn Univ 62 Sci AnenueHigh Tech Zone Zhengzhou 450001 Henan Peoples R China

Scientific and engineering applications rely on floating-point arithmetic to approximate real numbers. Due to the inherent rounding errors in floating-point numbers, error propagation during calculations can accumulate and lead to serious errors that may compromise the safety and reliability of the program. In theory, the most accurate method of error detection is to exhaustively search all possible floating-point inputs, but this is not feasible in practice due to the huge search space involved. Effectively and efficiently detecting maximum floating-point errors has been a challenge. To address this challenge, we design and implement an error detection tool for floating-point arithmetic expressions called HSED. It leverages modified mantissas under double precision floating-point types to simulate hierarchical searches from either half or single precision to double precision. Experimental results show that for 32 single-parameter arithmetic expressions in the FPBench benchmark test set, the error detection effects and performance of HSED are significantly better than the state-of-the-art error detection tools Herbie, S3FP and ATOMU. HSED outperforms Herbie, Herbie+, S3FP and ATOMU in 24, 19, 27 and 25 cases, respectively. The average time taken by Herbie, Herbie+, and S3FP is 1.82, 11.20, and 129.15 times longer than HSED, respectively.

关键词： floating-point arithmetic Error detection Dynamic analysis Hierarchical search

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The most precise computations using Euler's method in standard floating-point arithmetic applied to modelling of biological systems

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COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2013年第2期111卷 471-479页

作者： Kalinina, Elizabeth A. St Petersburg State Univ Fac Appl Math & Control Proc St Petersburg 199034 Russia

The explicit Euler's method is known to be very easy and effective in implementation for many applications. This article extends results previously obtained for the systems of linear differential equations with constant coefficients to arbitrary systems of ordinary differential equations. Optimal (providing minimum total error) step size is calculated at each step of Euler's method. Several examples of solving stiff systems are included. (C) 2013 Elsevier Ireland Ltd. All rights reserved.

关键词： Euler's method Optimal step size System of ordinary differential equations floating-point arithmetic Rounding errors Real-time systems

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A NEW APPROACH FOR BLOCK floating-point arithmetic IN RECURSIVE FILTERS

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS 1985年第7期32卷 719-722页

作者： WILLIAMSON, D SRIDHARAN, S MCCREA, PG SCH ELECT ENGN & COMP SCI DEPT COMP SCIKENSINGTONNSW 2033AUSTRALIA

An approach to block floating-point arithmetic in recursive second-order direct form digital filters is proposed. Used in conjunction with residue (or error) feedback, the method gives improved scaling and roundoff noise properties compared to an earlier approach [2]. It is conjectured from extensive simulation studies that the proposed secondorder structure is free from finite wordlength oscillations of all types.

关键词： floating-point arithmetic Digital filters Fixed-point arithmetic Computer science Dynamic range Feedback Digital arithmetic Hardware Control systems Adders

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Optimal controller and filter realizations using finite-precision, floating-point arithmetic

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INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 2005年第7期36卷 405-413页

作者： Whidborne, JF Gu, DW Wu, J Chen, S Cranfield Univ Dept Aerosp Sci Cranfield MK43 0AL Beds England Univ Leicester Dept Engn Leicester LE1 7RH Leics England Zhejiang Univ Inst Ind Proc Control Natl Key Lab Ind Control Technol Hangzhou 310027 Peoples R China Univ Southampton Dept Elect & Comp Sci Southampton SO17 1BJ Hants England

The problem of reducing the fragility of digital controllers and filters implemented using finite-precision, floating-point arithmetic is considered. floating-point arithmetic parameter uncertainty is multiplicative, unlike parameter uncertainty resulting from fixed-point arithmetic. Based on first-order eigenvalue sensitivity analysis, an upper bound on the eigenvalue perturbations is derived. Consequently, open-loop and closed-loop eigenvalue sensitivity measures are proposed. These measures are dependent upon the filter/controller realization. Problems of obtaining the optimal realization with respect to both the open-loop and the closed-loop eigenvalue sensitivity measures are posed. The problem for the open-loop case is completely solved. Solutions for the closed-loop case are obtained using non-linear programming. The problems are illustrated with a numerical example.

关键词： finite-precision arithmetic finite word length digital controller digital filter eigenvalue sensitivity floating-point arithmetic controller fragility

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Error in Ulps of the Multiplication or Division by a Correctly-Rounded Function or Constant in Binary floating-point arithmetic

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IEEE TRANSACTIONS ON EMERGING TOPICS IN COMPUTING 2024年第2期12卷 656-666页

作者： Brisebarre, Nicolas Muller, Jean-Michel Picot, Joris CNRS Lab LIP ENS Lyon F-69342 Lyon France ENS Lyon Lab LIP F-69342 Lyon France

Assume we use a binary floating-point arith-metic and that RN is the round-to-nearest function. Also assume that c is a constant or a real function of one or more variables, and that we have at our disposal a correctly rounded implementation of c, sayc=RN(c). For evaluat-ing xc (resp.x/c or c/x), the natural way is to replace it by RN (xc) (***(x/c) or RN(c/x)), that is, to call functioncand to perform a floating-point multiplica-tion or division. This can be generalized to the approxima-tion of n/dbyRN(n/d) and the approximation of ndbyRN(nd), wheren=RN(n) andd=RN(d),and n and d are functions for which we have at our disposal acorrectly rounded implementation. We discuss tight error bounds in ulps of such approximations. From our results, one immediately obtains tight error bounds for calculations such as x & lowast;pi,ln(2)/x,x/(y+z),(x+y)& lowast;z,x/sqrt(y),sqrt(x)/y,(x+y)(z+t),(x+y)/(z+t),(x+y)/(zt),etc. in floating-point arithmetic

关键词： floating-point arithmetic ulp numerical error correct rounding multiplication by a constant

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Enabling floating-point arithmetic in the Coq Proof Assistant

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JOURNAL OF AUTOMATED REASONING 2023年第4期67卷 1-30页

作者： Martin-Dorel, Erik Melquiond, Guillaume Roux, Pierre Univ Toulouse Toulouse INP IRIT CNRSUT3 Toulouse France Univ Paris Saclay LMF CNRS ENS Paris SaclayInria Gif Sur Yvette France Univ Toulouse ONERA DTIS Toulouse France

floating-point arithmetic is a well-known and extremely efficient way of performing approximate computations over the real numbers. Although it requires some careful considerations, floating-point numbers are nowadays routinely used to prove mathematical theorems. Numerical computations have been applied in the context of formal proofs too, as illustrated by the CoqInterval library. But these computations do not benefit from the powerful floating-point units available in modern processors, since they are emulated inside the logic of the formal system. This paper experiments with the use of hardware floating-point numbers for numerically intensive proofs verified by the Coq proof assistant. This gives rise to various questions regarding the formalization, the implementation, the usability, and the level of trust. This approach has been applied to the CoqInterval and ValidSDP libraries, which demonstrates a speedup of at least one order of magnitude.

关键词： Formal proof floating-point arithmetic Proof by computation

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Analysis and implementation of a novel leading zero anticipation algorithm for floating-point arithmetic units

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II-EXPRESS BRIEFS 2007年第8期54卷 685-689页

作者： Olivieri, Mauro Pappalardo, Francesco Smorfa, Simone Visalli, Giuseppe Univ Roma La Sapienza Dept Elect Engn I-00184 Rome Italy STMMicroelect I-95121 Catania Italy

Leading zero anticipation with error correction is a widely adopted technique in the implementation of high-speed IEEE-754-compliant floating-point units (FPUs), which are critical for area and power in multimedia-oriented systerns-on-chips. We investigated a novel LZA algorithm allowing us to remove error correction circuitry by reducing the error rate below a commonly accepted limit for image processing applications, which is not achieved by previous techniques. We embedded our technique into a complete FPU definitely obtaining both area saving and overall FPU latency reduction with respect to traditional designs.

关键词： CMOS VLSI floating-point unit (FPU) floating-point arithmetic leading zero anticipation (LZA)

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