Numerical stability

In numerical Analysis a branch of the Mathematical , the numerical stability is a property of the numerical algorithms. The precise definition of the stability depends on the context, but it relates to the exactitude of the results provided by an algorithm. It describes how the errors in the data input are propagated through an algorithm.

Sometimes a calculation can be carried out several manners, which all are algebraically equivalent and give the same result theoretically, but in practice they give different results because they have various levels of numerical stability. One of the common tasks of the numerical analysis is to try to find the algorithms most robust, i.e. having best numerical stability.

In a stable method, the errors remain tiny and the produced results are in conformity with those awaited. In an unstable method, the miscalculations are amplified by the treatment and deteriorate the end result. Unstable methods produce abhérants results quickly and are useless for the digital processing.

Opposite stability

Let us consider a problem solved by means of a numerical algorithm considered as a function $f$ which associates with the data $x$ the solution $y$. The real result noted $y^*$, will deviate in general from the exact solution. The leading causes are the errors rounding, the truncation errors and the data errors. The downstream error of an algorithm is the difference between the real result and the exact solution. The upstream error or opposite error is smallest $\backslash \; Delta\; x$ such as $f\; \backslash \; left\; (x+\; \backslash \; Delta\; X\; \backslash \; right)\; =y^*$; in other words, the upstream error indicates to us how the problem is really solved by the algorithm. The errors upstream and downstream are connected by the number condition: the downstream error at most has same the Order of magnitude as the condition number multiplied by the order of magnitude of the upstream error.

The algorithm is known as conversely stable or stable upstream if the upstream error is small enough for all the data $x$. Of course “small” is a relative term, and its definition will depend on the context. Often it is required that the error is of the same order as…, or only larger than… of some orders of magnitude, with a margin of a unit.

In much of situations, it is more natural to consider the relative error

$\backslash \; frac$

in the place of the absolute error $\backslash \; Delta\; x$.

Mixed stability

Stability of the numerical methods of resolution of the differential equations

References

• Nicholas J. Higham, Accuracy and Stability off Numerical Algorithms , Society off Industrial and Applied Mathematics, Philadelphia, 1996. ISBN 0-89871-355-2.

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