# 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 *x*_{0} 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 differential, integral and functional equations. 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.

## External links

- Jonathan Richard Shewchuk.
*An Introduction to the Conjugate Gradient Method Without the Agonizing Pain*, p. 6.