- Residual (numerical analysis)
Loosely speaking, a

**residual**is the error in a result. To be precise, suppose we want to find "x" such that: $f(x)=b.,$

Given an approximation of "x"

_{0}of "x", the residual is: $b\; -\; f(x\_0),$

whereas the error is

: $x\_0\; -\; x.,$

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

differential,

integral,

functional equations.For the approximation$~f\_\{\; m\; a\}~$ of solution $~f~$ of equation:$T(f)(x)=g(x)$ ,the residual can be either function: $~g(x)~\; -\; ~T(f\_\{\; m\; a\})(x)$or can be said to be maximum of the norm of this difference: $max\_\{xin\; mathcal\; X\}\; |g(x)-T(f\_\{\; m\; a\})(x)|$over the domain $mathcal\; X$, where the function $~f\_\{\; m\; a\}~$is expected to approximate the solution $~f~$,or some integral of a function of the difference, for example, $~int\_\{mathcal\; X\}\; |g(x)-T(f\_\{\; m\; a\})(x)|^2~\{\; m\; d\}\; x$ .In many cases, the smallness of the residual means that the approximation is close to the solution, i.e., : $~|f\_\{\; m\; a\}(x)\; -\; f(x)|\; ll\; 1~$. 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 solver s such as thegeneralized minimal residual method , which seeks solutions to equations by systematically minimizing the residual.**External links*** 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.

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**List of numerical analysis topics**— This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra … Wikipedia**Residual (mathematics)**— The word residual is used in a number of different senses in mathematics. See:* Errors and residuals in statistics * Residual (numerical analysis) ** Minimal residual method (MINRES) ** Generalized minimal residual method (GMRES) * Residual set,… … Wikipedia**Regression analysis**— In statistics, regression analysis is a collective name for techniques for the modeling and analysis of numerical data consisting of values of a dependent variable (response variable) and of one or more independent variables (explanatory… … Wikipedia**Generalized minimal residual method**— In mathematics, the generalized minimal residual method (usually abbreviated GMRES) is an iterative method for the numerical solution of a system of linear equations. The method approximates the solution by the vector in a Krylov subspace with… … Wikipedia**Least-squares spectral analysis**— (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. [cite book | title = Variable Stars As Essential Astrophysical Tools | author = Cafer Ibanoglu |… … Wikipedia**Principal component analysis**— PCA of a multivariate Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.878, 0.478) direction and of 1 in the orthogonal direction. The vectors shown are the eigenvectors of the covariance matrix scaled by… … Wikipedia**Outline of regression analysis**— In statistics, regression analysis includes any technique for learning about the relationship between one or more dependent variables Y and one or more independent variables X. The following outline is an overview and guide to the variety of… … Wikipedia**Wave equation analysis**— is a numerical method of analysis for the behavior of driven foundation piles. It predicts the pile capacity versus blow count relationship (bearing graph) and pile driving stress. The model mathematically represents the pile driving hammer and… … Wikipedia**List of mathematics articles (R)**— NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… … Wikipedia**Linear least squares (mathematics)**— This article is about the mathematics that underlie curve fitting using linear least squares. For statistical regression analysis using least squares, see linear regression. For linear regression on a single variable, see simple linear regression … Wikipedia