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.

Similar terminology is used dealing with differential, integral and functional equations. For the approximation ${\displaystyle ~f_{\rm {a}}~}$ of the solution ${\displaystyle ~f~}$ 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 ${\displaystyle {\mathcal {X}}}$, where the function ${\displaystyle ~f_{\rm {a}}~}$ is expected to approximate the solution ${\displaystyle ~f~}$, or some integral of a function of the difference, for example:

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