- 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 of "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 function
Similar terminology is used dealing with
functional equations.For the approximation of solution of equation: ,the residual can be either function: or can be said to be 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.
* Jonathan Richard Shewchuk. " [http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf An Introduction to the Conjugate Gradient Method Without the Agonizing Pain] ", p. 6.
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