Residual (numerical analysis)
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 exactly, we cannot compute the error but we can compute the residual.
Residual of the approximation of a function
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.
- Jonathan Richard Shewchuk. An Introduction to the Conjugate Gradient Method Without the Agonizing Pain, p. 6.