Loosely speaking, a residual is the error in a result.
[1] To be precise, suppose we want to find x such that Given an approximation x0 of x, the residual is that is, "what is left of the right hand side" after subtracting f(x0)" (thus, the name "residual": what is left, the rest).
Similar terminology is used dealing with differential, integral and functional equations.
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
, 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.