Validated high precision solution of second order initial value problem with Taylor model

• Autorzy: Stręk T.
• Konferencja: Symposium Vibrations in Physical Systems (XXI ; 26-29.05.2004 ; Poznań-Kierz, Polska)
• Języki publikacji: EN

Abstrakty

EN

The problem of reliability of computer computations is one of great concern to specialists in many areas of science and engineering. The notion of computing estimates of numerical error in computer simulations is not new. In recent years, considerable progress has been made in determining theoretical and computational techniques that aid to improve the reliability of results of simulations. An important advance in this area has been the recent discovery of methods to determine upper and lower bounds of local approximation error in any given simulation. Different from floating-point computations, interval arithmetic offers a simple mechanism to evaluate an enclosure of a function. Interval arithmetic is the arithmetic defined on sets of intervals, rather than sets of real numbers. The power of the interval arithmetic lay in implementation of interval arithmetic on computers. The fundamental problem in interval methods is computing the ranges of values of real function. The overestimation of the range of a given function by the interval arithmetic expression is strongly dependent on the arithmetic expression of the given function. The reason for this is based on the fact that interval arithmetic does not follow the same rules as the arithmetic for real numbers.

Słowa kluczowe

EN

numerical error; interval arithmetic; Taylor model

PL

błąd numeryczny; arytmetyka przedziałowa; model Taylora

• Wydawca: Poznan University of Technology. Institute of Applied Mechanics
• Czasopismo: Vibrations in Physical Systems
• Rocznik: 2004
• Tom: Vol. 21
• Strony: 357--360
• Opis fizyczny: Bibliogr. 9 poz., tab.

Twórcy

autor

Stręk T.

• Institute of Applied Mechanics, Poznan University of Technology

Bibliografia

• 1. J. T. Oden, T. Belytschko, I. Babuska and T.J.R. Hughes, Research directions in computational mechanics, Comput. Methods Appl. Mech. Engrg. 192 913-922 (2003).
• 2. R. E. Moore, Methods and applications of interval analysis, SIAM, Philadelphia (1979).
• 3. G. Alfeld and G. Mayer, Interval analysis: theory and applications. Journal of Computational and Applied Mathematics 121 421-464 (2000).
• 4. M. Berz and K. Makino, Verified Integration of ODEs and Flows using Differential Algebraic Methods on Higher-Order Taylor Models. Reliable Computing 4 361-369 (1998).
• 5. S. Wolfram, The Mathematica Book, 4th ed., Wolfram Media/Cambridge University Press, Champaign/Cambridge (1999).
• 6. K. Makino and M. Berz, Higher Order Verified Inclusions of Multidimensional Systems by Taylor Models. Nonlinear Analysis 47 3503-3514 (2001).
• 7. N.S. Nedialkov, K.R. Jackson and G.F. Corlis, Validated solutions of initial value problems for ordinary differential equations. Applied Mathematics and Computation 105 21-68 (1999).
• 8. Q. Lin and J.G. Rokne, Interval Approximation of Higher Order to the Ranges of Functions, Computers and Math. Applic. 31 (7) 101-109 (1996).
• 9. T. Strek, Verified inclusions of multivariate functions by Taylor method using symbolic algebra, Book of Abstract of the Annual Scientific Conference GAMM 2003, Abano-Terme, Italy (2003).

Uwagi

This work was supported by the State Committee for Scientific Research (KBN) under grant No 4 T11F 001 25 which is gratefully acknowledged.