Multistep Interval Methods of Nyström and Milne-Simpson Types
Poznań University of Technology, Institute of Computing Science
Piotrowo 3a, 60-965 Poznań, Poland
Adam Mickiewicz University, Faculty of Mathematics and Computer Science
Umultowska 87, 61-614 Poznań, Poland
e-mail: anmar@sol.put.poznan.pl
Received:
Rec. 24 January 2006
DOI: 10.12921/cmst.2007.13.01.23-29
OAI: oai:lib.psnc.pl:628
Abstract:
The paper is dealt with two kinds of multistep intervals methods which can be used to solve the initial value problem in the form of intervals containing all possible numerical errors. The interval methods of Nyström type are explicit, while the methods of Milne- Simpson are implicit. It appears that we can get two families of interval methods of the second kind. For both kinds of interval methods numerical examples are presented and compared with other interval multistep method considered in previous papers of the author.
Key words:
floating-point interval arithmetic, initial value problem, interval methods
References:
[1] J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, J. Wiley & Sons, Chichester 1987.
[2] K. Gajda, A. Marciniak, A. Marlewski, B. Szyszka, A Layout of an Object-Oriented System for Solving the Initial Value Problem by Interval Methods of Runge-Kutta Type [in Polish], Pro Dialog 8, 39-62 (1999).
[3] K. Gajda, A. Marciniak, B. Szyszka, Three- and Four-Stage Implicit Interval Methods of Runge-Kutta Type, Computational Methods in Science and Technology 6, 41-59 (2000).
[4] M. Jankowska, A. Marciniak, Implicit Interval Multistep Methods for Solving the Initial Value Problem, Computational Methods in Science and Technology 8(1), 17-30 (2002).
[5] M. Jankowska, A. Marciniak, On Explicit Interval Methods of Adams-Bashforth Type, Computational Methods in Science and Technology 8(2), 46-57 (2002).
[6] M. Jankowska, A. Marciniak, On Two Families of Implicit Interval Methods of Adams-Moulton Type, Computational Methods in Science and Technology 12(2), 109-113 (2006).
[7] S. A. Kalmykov, Ju. I. Šokin, E. Ch. Juldašev, On an Interval-Analytical Method of Second Order for Solving Ordinary Differential Equations [in Rusian], Izv. ANUzSSR, Ser. Fiz.-Mat. Nauk 3 (1976).
[8] S. A. Kalmykov, Ju. I. Šokin, E. Ch. Juldašev, Some Interval Methods for Solving Ordinary Differential Equations [in Russian], Èislennoje Metody Mehaniki Splošnoj Sredy 7(6), (1976).
[9] S. A. Kalmykov, Ju. I. Šokin, E. Ch. Juldašev, Solving Ordinary Differential Equations by Interval Methods [in Russian], Doklady AN SSSR, 230(6) (1976).
[10] F. Krückeberg, Ordinary Differential Equations, in: E. Hansen (Ed.), Topics in Interval Analysis, Oxford University Press, 91-97 (1969).
[11] A. Marciniak, Numerical Solutions of the N-body Problem, D. Reidel Publishing Co., Dordrecht 1985.
[12] A. Marciniak, 0,1, Pro Dialog 5, 55-82 (1997).
[13] A. Marciniak, B. Szyszka, One- and Two-Stage Implicit Interval Methods of Runge-Kutta Type, Computational Methods in Science and Technology 5, 53-65 (1999).
[14] A. Marciniak, Borland Delphi 5 Professional. Object Pascal [in Polish], NAKOM Publishers, Poznań 2000.
[15] A. Marciniak, Finding the Integration Interval for Interval Methods of Runge-Kutta Type in Floating-Point Interval Arithmetic, Pro Dialog 10, 35-45 (2000).
[16] A. Marciniak, B. Szyszka, On Representations of Coefficients in Implicit Interval Methods of Runge-Kutta Type, Computational Methods in Science and Technology 10(1), 57-71 (2004).
[17] A. Marciniak, Implicit Interval Methods for Solving the Initial Value Problem, Numerical Algorithms 37, 241-251 (2004).
[18] A. Marciniak, On Multistep Interval Methods for Solving the Initial Value Problem, Journal of Computational and Applied Mathematics (2006) (in press).
[19] R. E. Moore, Interval Analysis, Prentice Hall, 1966.
[20] B. Szyszka, Implicit Interval Methods of Runge-Kutta Type [in Polish], Ph. D. Thesis, Poznan University of Technology, 2003.
[21] Ju. I. Šokin, Interval Analysis [in Russian], Izdatel’stvo “Nauka”, Novosibirsk, 1981.
The paper is dealt with two kinds of multistep intervals methods which can be used to solve the initial value problem in the form of intervals containing all possible numerical errors. The interval methods of Nyström type are explicit, while the methods of Milne- Simpson are implicit. It appears that we can get two families of interval methods of the second kind. For both kinds of interval methods numerical examples are presented and compared with other interval multistep method considered in previous papers of the author.
Key words:
floating-point interval arithmetic, initial value problem, interval methods
References:
[1] J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, J. Wiley & Sons, Chichester 1987.
[2] K. Gajda, A. Marciniak, A. Marlewski, B. Szyszka, A Layout of an Object-Oriented System for Solving the Initial Value Problem by Interval Methods of Runge-Kutta Type [in Polish], Pro Dialog 8, 39-62 (1999).
[3] K. Gajda, A. Marciniak, B. Szyszka, Three- and Four-Stage Implicit Interval Methods of Runge-Kutta Type, Computational Methods in Science and Technology 6, 41-59 (2000).
[4] M. Jankowska, A. Marciniak, Implicit Interval Multistep Methods for Solving the Initial Value Problem, Computational Methods in Science and Technology 8(1), 17-30 (2002).
[5] M. Jankowska, A. Marciniak, On Explicit Interval Methods of Adams-Bashforth Type, Computational Methods in Science and Technology 8(2), 46-57 (2002).
[6] M. Jankowska, A. Marciniak, On Two Families of Implicit Interval Methods of Adams-Moulton Type, Computational Methods in Science and Technology 12(2), 109-113 (2006).
[7] S. A. Kalmykov, Ju. I. Šokin, E. Ch. Juldašev, On an Interval-Analytical Method of Second Order for Solving Ordinary Differential Equations [in Rusian], Izv. ANUzSSR, Ser. Fiz.-Mat. Nauk 3 (1976).
[8] S. A. Kalmykov, Ju. I. Šokin, E. Ch. Juldašev, Some Interval Methods for Solving Ordinary Differential Equations [in Russian], Èislennoje Metody Mehaniki Splošnoj Sredy 7(6), (1976).
[9] S. A. Kalmykov, Ju. I. Šokin, E. Ch. Juldašev, Solving Ordinary Differential Equations by Interval Methods [in Russian], Doklady AN SSSR, 230(6) (1976).
[10] F. Krückeberg, Ordinary Differential Equations, in: E. Hansen (Ed.), Topics in Interval Analysis, Oxford University Press, 91-97 (1969).
[11] A. Marciniak, Numerical Solutions of the N-body Problem, D. Reidel Publishing Co., Dordrecht 1985.
[12] A. Marciniak, 0,1, Pro Dialog 5, 55-82 (1997).
[13] A. Marciniak, B. Szyszka, One- and Two-Stage Implicit Interval Methods of Runge-Kutta Type, Computational Methods in Science and Technology 5, 53-65 (1999).
[14] A. Marciniak, Borland Delphi 5 Professional. Object Pascal [in Polish], NAKOM Publishers, Poznań 2000.
[15] A. Marciniak, Finding the Integration Interval for Interval Methods of Runge-Kutta Type in Floating-Point Interval Arithmetic, Pro Dialog 10, 35-45 (2000).
[16] A. Marciniak, B. Szyszka, On Representations of Coefficients in Implicit Interval Methods of Runge-Kutta Type, Computational Methods in Science and Technology 10(1), 57-71 (2004).
[17] A. Marciniak, Implicit Interval Methods for Solving the Initial Value Problem, Numerical Algorithms 37, 241-251 (2004).
[18] A. Marciniak, On Multistep Interval Methods for Solving the Initial Value Problem, Journal of Computational and Applied Mathematics (2006) (in press).
[19] R. E. Moore, Interval Analysis, Prentice Hall, 1966.
[20] B. Szyszka, Implicit Interval Methods of Runge-Kutta Type [in Polish], Ph. D. Thesis, Poznan University of Technology, 2003.
[21] Ju. I. Šokin, Interval Analysis [in Russian], Izdatel’stvo “Nauka”, Novosibirsk, 1981.
