Numerical stability

In the numerical analysis, a method is stable when it is insensitive to small perturbations of the data. In particular, this means that not too much affect rounding errors on the calculation. A distinction is made in the numerics this condition, stability and consistency, which are mutually closely related. Stability is a property of the algorithm and the condition of a property of the problem. The relationship between the condition of a problem and stability can be described as follows:

The mathematical problem it is a function of the input, and it is the numerical algorithm, and the disturbed input data. So one would like to estimate the following error:

By the triangle inequality holds:

This is denoted by the condition of the problem and the stability.

So the stability describes the robustness of the numerical method against disturbances in the input data, in particular this means that rounding errors do not accumulate and lead to disturbances in the solution. The quantification of the term differs, however, the problem and the used standard.

Stability and consistency depend usually together such that they are sometimes even with a small additional assumption is equivalent to convergence of the ( numerical ) solution against the analytical.

2.1 addition
2.2 Differential Equations 2.2.1 Ordinary Differential Equations
2.2.2 Partial Differential Equations

The two methods of analysis

Forward analysis

A process is stable if there is a constant, as is true that:

The relative condition of the problem and the machine precision respectively. quantifies the stability in the sense of forward analysis.

Backward analysis

The second common method of analysis is introduced by James Hardy Wilkinson backward analysis. Most of you know a useful upper bound for the inevitable relative input error ( problem dependent, the measurement error or even a rounding error to be). In order to estimate the error caused by the algorithm better, one expects him to assist in the backward analysis to an equivalent error in the input data of the problem, which is referred to as backward error. The formal definition of the backward error of the algorithm for the ( rounded ) Input Data ( ) is:

Where stands for domain.

The algorithm is stable reverse when the reverse relative error is less than all of the inevitable relative input errors. For some applications, it weakens this requirement now and still leaves the problem of a proper constant to the

To apply. Sometimes one is interested only in whether the relative backward error is at all limited.

One can show that backward stability implies forward stability.

Applications

Addition

Since one can show that the relative condition of the addition of two numbers in the case of extinction ( result is close to 0) can be arbitrarily bad, follows from the definition of the forward analysis that the addition of a numerical method ( the computer) is stable.

Differential equations

For numerical solvers for differential equations with initial or boundary values , or with right side trying to obtain an estimate of the developed solution of these input variables. In the sense of forward analysis, it is in this case constant.

Ordinary Differential Equations

Here the equivalence theorem of Lax, are after the zero - stability and consistency is equivalent to convergence of the method is valid.

Concrete method, the stability region is defined as the set of complex numbers for which the numerical method in solving the equation dahlquistschen test

With fixed step size is a monotonically decreasing sequence of approximations supplies.

The best case is when the stability region contains the entire left half-plane, then the method is called A-stable.

Partial Differential Equations

The standard method for stability analysis of numerical methods for partial differential equations is the Von Neumann stability analysis, which makes for linear problems necessary and sufficient statements, but only necessary for nonlinear problems.