Residual (numerical analysis)

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For other senses of the word "residual" in mathematics, see residual (mathematics).

Loosely speaking, a residual is the error in a result. To be precise, suppose we want to find x such that

$f(x)=b.\,$

Given an approximation of x₀ of x, the residual is

$b - f(x_0)\,$

whereas the error is

$x_0 - x.\,$

If we do not know x, we cannot compute the error but we can compute the residual.

[edit] Residual of the approximation of function

Similar terminology is used dealing with differential, integral, functional equations. For the approximation $~f_{\rm a}~$ of solution $~f~$ of equation

T (f)(x) = g (x)

the residual can be either function

$~g(x)~ - ~T(f_{\rm a})(x)$

or can be said to be maximum of the norm of this difference

$\max_{x\in \mathcal X} |g(x)-T(f_{\rm a})(x)|$

over the domain $\mathcal X$ , where the function $~f_{\rm a}~$ is expected to approximate the solution $~f~$ , or some integral of a function of the difference, for example, $~\int_{\mathcal X} |g(x)-T(f_{\rm a})(x)|^2~{\rm d} x$ .

In many cases, the smallness of the residual means that the approximation is close to the solution, i.e.,

$~|f_{\rm a}(x) - f(x)| \ll 1~$ .

In these cases, the initial equation is considered as well-posed; and the residual can be considered as a measure of deviation of the approximation from the exact solution.

[edit] Use or residuals

While one does not know the exact solution, one may look for the approximation with small residual.

Residuals appear in many areas in mathematics, from iterative solvers such as the generalized minimal residual method, which seeks solutions to equations by systematically minimizing the residual.