Minimum asymptotic error of algorithms for solving ODE
This paper extends previous results of the author [Numer. Math. 45, 93- 104 (1984; Zbl 0527.65055) and J. Complexity 3, 451-465 (1987; Zbl 0643.65033)] concerned with the error of some algorithms for the numerical solution ordinary differential operations. Here the error is to be understood as the largest value of the difference between the real and the computed solution of the initial value problem \(y'=f(y,t)\), \(y(t_ 0)=y_ 0\), \(t\in [0,c]\) when f belongs to a space of functions with r continuous bounded derivatives. First, assuming that the algorithms are dependent on f through the values of n linear continuous functionals at f, it is proved that the error cannot converge to zero faster than \(n^{-r}\) as \(n\to +\infty\) and this minimal error is achieved by the Taylor algorithm. Finally, for arbitrary linear functionals, the optimal error bounds are of order \(n^{-(r+1)}\) when \(n\to +\infty\).
scientific article; zbMATH DE number 4082871