Residual (numerical analysis)

As residuum is called in mathematics, especially in the numerical analysis, the deviation from the desired result, which arises when approximate solutions are used in an equation. Suppose there is given a function and you want to find an x such that

With an approximation to the residual

The error, however,

The error is usually unknown, since x is unknown, which is why this can not be used as a termination criterion in a numerical method. The residue is, however, always available.

If the residue is small, the following, in many cases, that the approximation is close to the solution, that is,

In these cases, the equation to be solved is seen as well provided and the residue can be considered a measure of the deviation of the approximation from the exact solution. For linear systems of equations, the norm of the error and the residual can differ by a factor of condition.

Residuum of an approximation to a function

The concept of the residual for differential, integral and functional equations is used analog in which, instead of a number x is a function sought that an equation

Met. For an approximation to the residue, the function

As a measure of the quality of the approximation can then, for example, the maximum norm of the difference

Over the range in which the function is to approximate the solution, or as an integral

Be selected.