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A NOTE ON INEXACT INNER PRODUCTS IN GMRES

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2022年第3期43卷 1406-1422页

作者： Gratton, Serge Simon, Ehouarn Titley-peloquin, David Toint, Philippe L. INPT IRIT ENSEEIHT BP 7122 F-31071 Toulouse 7 France McGill Univ Dept Bioresource Engn Sainte Anne De Bellevue PQ H9X 3V9 Canada Univ Namur Namur Ctr Complex Syst naXys B-5000 Namur Belgium

We show to what extent the accuracy of the inner products computed in the GMRES iterative solver can be reduced as the iterations proceed without affecting the convergence rate or final accuracy achieved by the iterates. We bound the loss of orthogonality in GMRES with inexact inner products. We use this result to bound the ratio of the residual norm in inexact GMRES to the residual norm in exact GMRES and give a condition under which this ratio remains close to 1. We illustrate our results with examples in variable floating-point arithmetic.

关键词： Arnoldi algorithm GMRES algorithm inexact inner products variable precision arithmetic

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Numerical aspects of integration in semi-closed option pricing formulas for stochastic volatility jump diffusion models

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2020年第6期97卷 1268-1292页

作者： Danek, Josef Pospisil, Jan Univ West Bohemia Fac Appl Sci NTIS Univ 8 Plzen 30100 Czech Republic

In mathematical finance, a process of calibrating stochastic volatility (SV) option pricing models to real market data involves a numerical calculation of integrals that depend on several model parameters. This optimization task consists of large number of integral evaluations with high precision and low computational time requirements. However, for some model parameters, many numerical quadrature algorithms fail to meet these requirements. We can observe an enormous increase in function evaluations, serious precision problems and a significant increase of computational time. In this paper, we numerically analyse these problems and show that they are especially caused by inaccurately evaluated integrands. We propose a fast regime switching algorithm that tells if it is sufficient to evaluate the integrand in standard double arithmetic or if a higher precision arithmetic has to be used. We compare and recommend numerical quadratures for typical SV models and different parameter values, especially for problematic cases.

关键词： variable precision arithmetic numerical integration adaptive quadrature option pricing stochastic volatility models

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MATLAB implementation for evaluation of measurements by the generalized method of least squares

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MEASUREMENT 2018年 114卷 218-225页

作者： Solaguren-Beascoa Fernandez, M. Univ Burgos Escuela Politecn Super Struct Integr Grp Avda Cantabria S-N Burgos 09006 Spain

The method of Least Squares is one of the fundamental tools in scientific investigation. In 2002, Lars Nielsen described a technique for applying it in its more generalized form, fully compatible with the principles for evaluation of measurement uncertainty. Despite the importance of this contribution, its application has hardly been widespread, probably due to the technicalities of its implementation. Those difficulties are found in an iteration process that contains a near-singular system of equations. In this paper, the work of Nielsen has been reformulated in accordance with the notation of the ISO 'Guide to the expression of uncertainty in measurement', analyzed in detail to solve the implementation difficulties, completed with an adequate convergence criterion, and implemented in the MATLAB mathematical program. The result is a practical tool that will allow the application of the general least-squares method in accordance with the international requirements of expressing the measurements with their uncertainties.

关键词： Least-squares method Uncertainty Metrology Lagrange multipliers Ill-conditioning variable precision arithmetic

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Long term evolution of distant retrograde orbits in the Earth-Moon system

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ASTROPHYSICS AND SPACE SCIENCE 2017年第9期362卷 176-176页

作者： Bezrouk, Collin Parker, Jeffrey S. Univ Colorado Colorado Ctr Astrodynam Res ECNT 320 UCB 431 Boulder CO 80309 USA

This work studies the evolution of several Distant Retrograde Orbits (DROs) of varying size in the Earth-Moon system over durations up to tens of millennia. This analysis is relevant for missions requiring a completely hands off, long duration quarantine orbit, such as a Mars Sample Return mission or the Asteroid Redirect Mission. Four DROs are selected from four stable size regions and are propagated for up to 30,000 years with an integrator that uses extended precision arithmetic techniques and a high fidelity dynamical model. The evolution of the orbit's size, shape, orientation, period, out-of-plane amplitude, and Jacobi constant are tracked. It has been found that small DROs, with minor axis amplitudes of approximately 45,000 km or less decay in size and period largely due to the Moon's solid tides. Larger DROs (62,000 km and up) are more influenced by the gravity of bodies external to the Earth-Moon system, and remain bound to the Moon for significantly less time.

关键词： Distant retrograde orbit Dtability variable precision arithmetic

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The use of configurable computing for computational kernels in scientific simulations

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FUTURE GENERATION COMPUTER SYSTEMS-THE INTERNATIONAL JOURNAL OF ESCIENCE 2006年第1-2期22卷 67-79页

作者： Jones, M Nakad, Z Plassmann, P Yi, YH Virginia Polytech Inst & State Univ Bradley Dept Elect & Comp Engn Virginia Tech Blacksburg VA 24061 USA Penn State Univ Dept Comp Sci & Engn University Pk PA 16802 USA

In many scientific simulation codes, the bulk of the floating-point arithmetic required is done by a small number of compact computational kernels. In this paper, we explore the potential use of configurable computers to instantiate the hardware required for such kernels and, thus, improve their performance. We present algorithms and analysis for two such kernels: fast, problem-specific multipliers and the efficient evaluation of Taylor series. A novel aspect of the algorithm for Taylor series evaluation is that it takes advantage of the variable precision arithmetic available to a configurable computer. Experimental results obtained on a Xilinx field-programmable gate array (FPGA) are presented for the proposed algorithms. (c) 2004 Published by Elsevier B.V.

关键词： configurable computing FPGA variable precision arithmetic polynomial evaluation Taylor series approximation

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The use of configurable computing for computational kernels in scientific simulations

The use of configurable computing for computational kernels ...

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International Multi-Conference in Computer Science and Computer Engineering

关键词： configurable computing FPGA variable precision arithmetic polynomial evaluation Taylor series approximation

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The use of configurable computing in scientific simulations

The use of configurable computing in scientific simulations

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International Conference on Parallel and Distributed Processing Techniques and Applications

作者： Jones, M Park, J Yi, Y Plassmann, P Virginia Tech Dept Elect & Comp Engn Blacksburg VA 24061 USA

ISBN: (纸本)1892512459

A common feature of many simulation codes used in scientific computation is that the bulk of the floating point arithmetic is done by a small number of computational kernels. In this paper we explore the potential of using configurable computers to instantiate the hardware required for such kernels. We present an algorithm and analysis for an efficient implementation of the Taylor series. A novel aspect of this algorithm is that it takes advantage of variable precision arithmetic in the computation of the Taylor series. We present experimental results obtained on a Xilinx field-programmable gate array (FPGA) for the proposed algorithm.

关键词： configurable computing FPGA variable precision arithmetic polynomial evaluation Taylor series

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A Machine and Configuration Independent Fortran: Portable Fortran {PFortran}

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IEEE Transactions on Software Engineering 1975年第1期SE-1卷 111-124页

作者： Whitten, Douglas E. demaine, Paul A. D. Department of Computer Science Pennsylvania State University University Park Pa. 16802 United States

This paper describes the design of a programming language suitable for writing portable, machine independent programs. Based on Fortran, and implemented at the lowest level by means of subprograms, this language (PFortran) provides features which allow programs to be designed so as to be rather insensitive to computer architecture (byte versus word addressable) or to main memory size. The complete syntax of PFortran is currently being developed, and a PFortran-to-Fortran compiler is being written. This paper describes a PFortran program after compilation: as a pure Fortran program utilizing a special library of service subprograms. Fortran includes the features of: machine independent I/O, data compression/decompression, virtual data arrays (having generalized subscript forms), variable precision arithmetic, and a programmer-controlled data unit called the kernel which acts like a variable-sized byte or word. The PFortran compiler will be written in Fortran. Its output will be directly compilable using a wide range of existing standard Fortran compilers for specific machines. The support library routines are written in Fortran and assembly language, and designed for rapid conversion to most machines. Copyright © 1975 by The Institute of Electrical and Electronics Engineers, Inc.

关键词： Data structures Fortran machine independent programming program portability variable precision arithmetic virtual arrays

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