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

that is, "what is left of the right hand side" after subtracting f(x₀)" (thus, the name "residual": what is left, the rest). On the other hand, the error is

If the exact value of x is not known, the residual can be computed, whereas the error cannot.

Similar terminology is used dealing with differential, integral and functional equations. For the approximation ${\displaystyle f_{\text{a}}}$ 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 ${\displaystyle f_{\text{a}}}$ is expected to approximate the solution ${\displaystyle f}$,