Loosely speaking, a residual is the error in a result. To be precise, suppose we want to find x such that
Given an approximation x0 of x, the residual is
whereas the error is
If we do not know x, we cannot compute the error but we can compute the residual.
Residual of the approximation of a function
Similar terminology is used dealing with
For the approximation
of the solution
of the equation
the residual can either be the function
or can be said to be the maximum of the norm of this difference
over the domain , where the function
is expected to approximate the solution ,
or some integral of a function of the difference, for example:
In many cases, the smallness of the residual means that the approximation is close to the solution, i.e.,
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.
Use of 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.