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Numerische Mathematik

, Volume 70, Issue 1, pp 119–128 | Cite as

Error analysis of a randomized numerical method

  • Gilbert Stengle

Summary.

We analyze a randomized algorithm for solving the initial value problem for the finite dimensional system\( {dy \over dt} = f (t, y)\; ; \quad y (t_0) = y_0 \) on a finite interval under the hypothesis that \(f\) is smooth in\(y\) but no more than bounded and measurable in \(t\). The algorithm is a representative member of an infinite family of methods akin to the Runge-Kutta family. It generates a sequence \(Y_k\) by iterating the single step formula\( Y_1 = Y_0 + {h \over 2 p} \sum_{j=1}^ p \{ f (U_j, Y_0 + hf (u_j, Y_0)) + f (u_j, Y_k)\} $$ $$ U_j = \max (T_{1j}, T_{2j})\; , \quad u_j = \min (T_{1j}, T_{2j}) \) where for each step \(\{ T_{1j}\} \) and \(\{ T_{2j}\}\) are fresh\(p\) -fold random samples of the uniform distribution on \([t_0,t_0+h]\). We analyze the resulting random estimator \(Y\) of \(y\) by giving a weak law for its error that, with probability \(1 - \beta\), gives error bounds in terms of the parameters \(\beta\), \(h\) and \(p\).

Mathematics Subject Classification (1991): 34A50, 65C05, 621.20 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Gilbert Stengle
    • 1
  1. 1.Department of Mathematics, Lehigh University, Bethlehem, PA 18015-3174, USA US

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