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A stochastic arithmetic for reliable scientific computation

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  • Vignes, J.

Abstract

After having reviewed the CESTAC (Contrôle et Estimation STochastique des Arrondis de Calculs) method, which uses a probabilistic approach to round-off error analysis, a new arithmetic called stochastic arithmetic is presented in this paper. This new arithmetic covers a large part of the properties of exact arithmetic, properties which are lost in the usual floating-point arithmetic. The CADNA (Control of Accuracy and Debugging for Numerical Applications) library is a tool for automatic implementation of stochastic arithmetic in any fortran or ADA program. Consequently, by the use of the CADNA library, it is possible during the run of a program, to detect the numerical instabilities, to stop correctly any iterative process, to evaluate the optimal step or the optimal mesh in approximate methods, to evaluate the accuracy of any result provided by the computer. In short, the stochastic arithmetic, i.e., the use of the CADNA library, serves to validate the results provided by a computer, and to assure the user of the reliability of scientific computations.

Suggested Citation

  • Vignes, J., 1993. "A stochastic arithmetic for reliable scientific computation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 35(3), pages 233-261.
    • Handle: RePEc:eee:matcom:v:35:y:1993:i:3:p:233-261
    DOI: 10.1016/0378-4754(93)90003-D
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    References listed on IDEAS

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    1. Vignes, J., 1988. "Review on stochastic approach to round-off error analysis and its applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 30(6), pages 481-491.
    2. Alt, René, 1988. "Floating-point error propagation in iterative methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 30(6), pages 505-517.
    3. Vignes, Jean, 1984. "An efficient implementation of optimization algorithms," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 26(3), pages 243-256.
    4. Vignes, J., 1978. "New methods for evaluating the validity of the results of mathematical computations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 20(4), pages 227-249.
    5. Vergnes, J., 1980. "Determination d'un pas optimum d'integration pour la methode de Simpson," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 22(3), pages 177-188.
    6. Vergnes, J. & Dumontet, J., 1979. "Finding an optimal partition for a numerical integration using the trapezoidal rule," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 21(2), pages 231-241.
    7. Tolla, Pierre, 1983. "Linear and non-linear programming software validity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 25(1), pages 39-42.
    8. Bois, P. & Vignes, J., 1980. "A software for evaluating local accuracy in the fourier transform," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 22(2), pages 141-150.
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    1. Albertsen, Niels Christian & Chesneaux, Jean-Marie & Christiansen, Søren & Wirgin, Armand, 1999. "Comparison of four software packages applied to a scattering problem1Professor Ralph E. Kleinman, University of Delaware, USA, in memoriam.1," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 48(3), pages 307-317.
    2. Jézéquel, F. & Rico, F. & Chesneaux, J.-M. & Charikhi, M., 2006. "Reliable computation of a multiple integral involved in the neutron star theory," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 71(1), pages 44-61.
    3. Alt, R. & Lamotte, J.-L., 2001. "Experiments on the evaluation of functional ranges using a random interval arithmetic," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 56(1), pages 17-34.
    4. Samad Noeiaghdam & Sanda Micula, 2021. "Dynamical Strategy to Control the Accuracy of the Nonlinear Bio-Mathematical Model of Malaria Infection," Mathematics, MDPI, vol. 9(9), pages 1-24, May.
    5. Guilain, S. & Vignes, J., 1994. "Validation of numerical software results — Application to the computation of apparent heat release in direct-injection diesel engines," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 37(1), pages 73-92.
    6. Samad Noeiaghdam & Sanda Micula, 2021. "A Novel Method for Solving Second Kind Volterra Integral Equations with Discontinuous Kernel," Mathematics, MDPI, vol. 9(17), pages 1-12, September.
    7. Samad Noeiaghdam & Sanda Micula & Juan J. Nieto, 2021. "A Novel Technique to Control the Accuracy of a Nonlinear Fractional Order Model of COVID-19: Application of the CESTAC Method and the CADNA Library," Mathematics, MDPI, vol. 9(12), pages 1-26, June.
    8. Samad Noeiaghdam & Denis Sidorov & Abdul-Majid Wazwaz & Nikolai Sidorov & Valery Sizikov, 2021. "The Numerical Validation of the Adomian Decomposition Method for Solving Volterra Integral Equation with Discontinuous Kernels Using the CESTAC Method," Mathematics, MDPI, vol. 9(3), pages 1-15, January.
    9. Samad Noeiaghdam & Denis Sidorov & Alyona Zamyshlyaeva & Aleksandr Tynda & Aliona Dreglea, 2020. "A Valid Dynamical Control on the Reverse Osmosis System Using the CESTAC Method," Mathematics, MDPI, vol. 9(1), pages 1-17, December.
    10. Samad Noeiaghdam & Aliona Dreglea & Hüseyin Işık & Muhammad Suleman, 2021. "A Comparative Study between Discrete Stochastic Arithmetic and Floating-Point Arithmetic to Validate the Results of Fractional Order Model of Malaria Infection," Mathematics, MDPI, vol. 9(12), pages 1-17, June.
    11. Salkuyeh, Davod Khojasteh & Toutounian, Faezeh & Yazdi, Hamed Shariat, 2008. "A procedure with stepsize control for solving n one-dimensional IVPs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(2), pages 167-176.

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