Stepsize Control of the Finite Difference Method for Solving Ordinary Differential Equations
An important task in solving second order linear ordinary differential equations by the finite difference is to choose a suitable stepsize h. In this paper, by using the stochastic arithmetic, the CESTAC method and the CADNA library we present a procedure to estimate the optimal stepsize hopt, the stepsize which minimizes the global error consisting of truncation and round-off error.
[1] S. Abbasbandy and M. A. Fariborzi Araghi, A stochastic scheme for
solving definite integrals, Appl. Numer. Math., 55 (2005) 125-136.
[2] J. M. Chesneaux, L-arithm'etique stochastique et le logiciel CADNA,
Habilitation `a diriger des recherches, Universit'e Pierre et Marie Curie,
Paris, 1995.
[3] J. M. Chesneaux, CADNA: An ADA tool for round-off errors analysis
and for numerical debugging, In ADA in Aerospace, Barcelone, Spain,
December 1990.
[4] J. M. Chesneaux and F. J'ez'equel, Dynamical Control of Computations
Using the Trapezodial and Simpson-s rules, J. Universal Comput. Sci., 4
(1998) 2-10.
[5] J.M. Chesneaux, Study of the computing accuracy by using probabilistic
approach, in: C. Ullrich (Ed.), Contribution to Computer Arithmetic and
Self-Validating Numerical Methods, IMACS, New Brunswick, NJ, 1990.
[6] J. M. Chesneaux and J. Vignes, Sur la robustesse de la m'ethode CESTAC,
C. R. Acad. Sci. Paris, S'er. I Math., 307 (1988) 855-860.
[7] J. M. Chesneaux, Stochastic arithmetic properties, IMACS Comput. Appl.
Math. (1992) 81-91.
[8] J. M. Chesneaux and J. Vignes, Les fondements de l-arithm'etique
stochastique, C. R. Acad. Sci. Paris, S'er. I Math., 315 (1992) 1435-1440.
[9] F. J'ez'equel, A dynamical strategy for approximation methods, C. R.
M'ecanique, 334 (2006) 362-367.
[10] F. J'ez'equel and J. M. Chesneaux, CADNA: a library for estimating
round-off error propagation, Computer Physics Communications, 178
(2008) 933-955.
[11] F. J'ez'equel, Contrˆole dynamique de m'ethodes d-approximation,
Habilitation 'a diriger des recherches, Universit'e Pierre et
Marie Curie, Paris, February 2005, Available at: http://wwwanp.
lip6.fr/Ôê╝jezequel/HDR.html.
[12] J. M. Ortega, Numerical Analysis, A second course, SIAM, New York,
1990.
[13] A. Quarteroni, R. Sacco and F. Saleri, Numerical mathematics, Springer-
Verlag, New York, 2000.
[14] D. K. Salkuyeh, F. Toutounian and H. S. Yazdi, A procedure with
stepsize control for solving n one-dimensional IVPs, Mathematics and
Computers in Simulation, 79 (2008) 167-176.
[15] The CADNA library, URL address: http://www.lip6.fr/cadna.
[16] J. Vignes, Discrete stochastic arithmetic for validating results of numerical
software, Numerical Algorithms, 37 (2004) 377-390.
[17] J. Vignes, New methods for evaluating the validity of the results of
mathematical computations, Math. Comp. Simul., 20 (1978) 227-249.
[18] J. Vignes and M. La Porte, Error analysis in computing, in: Information
Processing 1974, North-Holand, (1974) 610-614.
[19] J. Vignes, A stochastic arithmetic for reliable scientific computation,
Math. Comp. Simul., 35 (1993) 233-261.
[20] J. Vignes, New methods for evaluating the validity of the results of
mathematical computations, Math. Comp. Simul., 20(1978) 227-249.
[21] J. Vignes, A stochastic approach to the analysis of round-off error
propagation. A survey of the CESTAC method, in: Proc. 2nd Real
Numbers and Computers Conference, Marseille, France, (1996) 233-251.
[22] J. Vignes, Z'ero math'ematique et z'ero informatique, C. R. Acad. Sci.
Paris S'er. I Math., 303 (1986) 997-1000.
[1] S. Abbasbandy and M. A. Fariborzi Araghi, A stochastic scheme for
solving definite integrals, Appl. Numer. Math., 55 (2005) 125-136.
[2] J. M. Chesneaux, L-arithm'etique stochastique et le logiciel CADNA,
Habilitation `a diriger des recherches, Universit'e Pierre et Marie Curie,
Paris, 1995.
[3] J. M. Chesneaux, CADNA: An ADA tool for round-off errors analysis
and for numerical debugging, In ADA in Aerospace, Barcelone, Spain,
December 1990.
[4] J. M. Chesneaux and F. J'ez'equel, Dynamical Control of Computations
Using the Trapezodial and Simpson-s rules, J. Universal Comput. Sci., 4
(1998) 2-10.
[5] J.M. Chesneaux, Study of the computing accuracy by using probabilistic
approach, in: C. Ullrich (Ed.), Contribution to Computer Arithmetic and
Self-Validating Numerical Methods, IMACS, New Brunswick, NJ, 1990.
[6] J. M. Chesneaux and J. Vignes, Sur la robustesse de la m'ethode CESTAC,
C. R. Acad. Sci. Paris, S'er. I Math., 307 (1988) 855-860.
[7] J. M. Chesneaux, Stochastic arithmetic properties, IMACS Comput. Appl.
Math. (1992) 81-91.
[8] J. M. Chesneaux and J. Vignes, Les fondements de l-arithm'etique
stochastique, C. R. Acad. Sci. Paris, S'er. I Math., 315 (1992) 1435-1440.
[9] F. J'ez'equel, A dynamical strategy for approximation methods, C. R.
M'ecanique, 334 (2006) 362-367.
[10] F. J'ez'equel and J. M. Chesneaux, CADNA: a library for estimating
round-off error propagation, Computer Physics Communications, 178
(2008) 933-955.
[11] F. J'ez'equel, Contrˆole dynamique de m'ethodes d-approximation,
Habilitation 'a diriger des recherches, Universit'e Pierre et
Marie Curie, Paris, February 2005, Available at: http://wwwanp.
lip6.fr/Ôê╝jezequel/HDR.html.
[12] J. M. Ortega, Numerical Analysis, A second course, SIAM, New York,
1990.
[13] A. Quarteroni, R. Sacco and F. Saleri, Numerical mathematics, Springer-
Verlag, New York, 2000.
[14] D. K. Salkuyeh, F. Toutounian and H. S. Yazdi, A procedure with
stepsize control for solving n one-dimensional IVPs, Mathematics and
Computers in Simulation, 79 (2008) 167-176.
[15] The CADNA library, URL address: http://www.lip6.fr/cadna.
[16] J. Vignes, Discrete stochastic arithmetic for validating results of numerical
software, Numerical Algorithms, 37 (2004) 377-390.
[17] J. Vignes, New methods for evaluating the validity of the results of
mathematical computations, Math. Comp. Simul., 20 (1978) 227-249.
[18] J. Vignes and M. La Porte, Error analysis in computing, in: Information
Processing 1974, North-Holand, (1974) 610-614.
[19] J. Vignes, A stochastic arithmetic for reliable scientific computation,
Math. Comp. Simul., 35 (1993) 233-261.
[20] J. Vignes, New methods for evaluating the validity of the results of
mathematical computations, Math. Comp. Simul., 20(1978) 227-249.
[21] J. Vignes, A stochastic approach to the analysis of round-off error
propagation. A survey of the CESTAC method, in: Proc. 2nd Real
Numbers and Computers Conference, Marseille, France, (1996) 233-251.
[22] J. Vignes, Z'ero math'ematique et z'ero informatique, C. R. Acad. Sci.
Paris S'er. I Math., 303 (1986) 997-1000.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:60475", author = "Davod Khojasteh Salkuyeh", title = "Stepsize Control of the Finite Difference Method for Solving Ordinary Differential Equations", abstract = "An important task in solving second order linear ordinary differential equations by the finite difference is to choose a suitable stepsize h. In this paper, by using the stochastic arithmetic, the CESTAC method and the CADNA library we present a procedure to estimate the optimal stepsize hopt, the stepsize which minimizes the global error consisting of truncation and round-off error.
", keywords = "Ordinary differential equations, optimal stepsize, error, stochastic arithmetic, CESTAC, CADNA.", volume = "3", number = "9", pages = "712-5", }
