Stability of a computational algorithm

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A partially resolving operator $L_m^h$, uniformly bounded in $h$ and $m$, describing the succession of steps in the computational algorithm for solving the equation


for example, a grid equation with step $h$ (cf. Closure of a computational algorithm). Stability of a computational algorithm guarantees weak influence of computational errors on the result of the calculation. However, the possibility is not excluded that the quantity $p(h)=\sup\|L_m^h\|$ grows comparatively slowly and a corresponding strengthening of the influence of computational errors for $h\to0$ remains practically admissible. The concept of stability of a computational algorithm has been made concrete in applications to grid-projection methods (cf. [4]) and in applications to iterative methods (cf. [6]). There are also other definitions of the stability of a computational algorithm (cf. e.g. [1], [3]).


[1] I. [I. Babushka] Babuška, M. Práger, E. Vitásek, "Numerical processes in differential equations" , Wiley (1966)
[2] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)
[3] M.K. Gavurin, "Lectures on computing methods" , Moscow (1971) (In Russian)
[4] G.I. Marchuk, V.I. Agoshkov, "Introduction to grid-projection methods" , Moscow (1981) (In Russian)
[5] A.A. Samarskii, A.V. Gulin, "Stability of difference schemes" , Moscow (1973) (In Russian)
[6] A.A. Samarskii, E.S. Nikolaev, "Numerical methods for grid equations" , 1–2 , Birkhäuser (1989) (Translated from Russian)
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Stability of a computational algorithm. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.F. Shapkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article