Loosely speaking, a residual is the

error An error (from the Latin ''error'', meaning "wandering") is an action which is inaccurate or incorrect. In some usages, an error is synonymous with a mistake. The etymology derives from the Latin term 'errare', meaning 'to stray'. In statistics ...

in a result. To be precise, suppose we want to find ''x'' such that :

f(x)=b.

Given an approximation ''x''₀ of ''x'', the residual is :

b - f(x_0)

that is, "what is left of the right hand side" after subtracting ''f''(''x''₀)" (thus, the name "residual": what is left, the rest). On the other hand, the error is :

x - x_0

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

Residual of the approximation of a function

Similar terminology is used dealing with differential,

integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...

and

functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...

s. For the approximation

f_\text

of the solution

f

of the equation :

T(f)(x)=g(x) \, ,

the residual can either be the function :

~g(x)~ - ~T(f_\text)(x)

or can be said to be the maximum of the norm of this difference :

\max_ , g(x)-T(f_\text)(x),

over the domain

\mathcal X

, where the function

f_\text

is expected to approximate the solution

f

, or some integral of a function of the difference, for example: :

\int_ , g(x)-T(f_\text)(x), ^2~ \mathrm dx.

In many cases, the smallness of the residual means that the approximation is close to the solution, i.e., :

\left, \frac\ \ll  1.

In these cases, the initial equation is considered as

well-posed The mathematical term well-posed problem stems from a definition given by 20th-century French mathematician Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that: # a solution exists, # the sol ...

; and the residual can be considered as a measure of deviation of the approximation from the exact solution.

Use of residuals

When one does not know the exact solution, one may look for the approximation with small residual. Residuals appear in many areas in mathematics, including

iterative solver In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pre ...

s such as the

generalized minimal residual method In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace wit ...

, which seeks solutions to equations by systematically minimizing the residual.

References

{{Reflist Numerical analysis