The use of integral information in the solution of a two-point boundary value problem

• Autorzy: Drwięga T.
• Języki publikacji: EN

Abstrakty

EN

We study the worst-case ε-complexity of a two-point boundary value problem u″(x) = ƒ(x) u (x), x ∈ [0,T], u(0) = c, u′ (T) = 0, where c,T ∈ R (c ≠ 0, T > 0) and ƒ is a nonnegative function with r (r ≥ 0) continuous bounded derivatives. We prove an upper bound on the complexity for linear information showing that a speed-up by two orders of magnitude can be obtained compared to standard information. We define an algorithm based on integral information and analyze its error, which provides an upper bound on the ε-complexity.

Słowa kluczowe

EN

boundary value problem; complexity; worst case setting; linear information

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2007
• Tom: Vol. 27, no. 2
• Strony: 205--220
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Drwięga T.

• AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Cracow, Poland,

Bibliografia

• [1] P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston 1982.
• [2] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962.
• [3] B. Kacewicz, How to increase the order to get minimal-error algorithms for systems of ODE, Numer. Math. 45 (1984), 1, 93–104.
• [4] B. Kacewicz, Optimal solution of some two-point boundary value problem, Numerical analysis and mathematical methods, Banach Center Publications 24 (1990), 241–256, PWN, Warsaw.
• [5] B. Kacewicz, Almost optimal solution of initial-value problems by randomized and quantum algorithms, J. Complexity 22 (2006), 676–690.
• [6] J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, PWN, Warsaw 1987 (Polish edition).
• [7] A.G. Werschulz, Optimal solution for a problem of optimal control, J. Complexity 5 (1989), 144–181.
• [8] J.H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965.
• [9] J.F. Traub, H.Wozniakowski, G.Wasilkowski, Information-Based Complexity, Academic Press, New York, 1988.