Residual (numerical mathematics)

This article covers the numerical point of view. For a statistical perspective, see Disturbance and Residual .

In numerical mathematics, the residual is the deviation from the desired result that arises when approximate solutions are used in an equation . Suppose there is a function and you want to find one such that ${\displaystyle f}$ ${\displaystyle x}$

${\displaystyle f(x)=b.}$

With an approximation to is the residual ${\displaystyle x_{0}}$ ${\displaystyle x}$ ${\displaystyle r}$

${\displaystyle r=b-f(x_{0}),}$

the mistake, however

${\displaystyle x_{0}-x.}$

The error is usually unknown, since x is unknown, which is why it cannot be used as a termination criterion in a numerical procedure. The residual, however, is always available.

In many cases, when the residual is small, it follows that the approximation is close to the solution, that is

${\displaystyle {\frac {\|x_{0}-x\|}{\|x\|}}\ll 1.~}$

In these cases the equation to be solved is considered to be well set and the residual can be viewed as a measure of the deviation of the approximation from the exact solution. In linear systems of equations , the norm of the relative error and the norm of the relative residual can differ by the factor of the condition , i.e.

${\displaystyle {\frac {\|x_{0}-x\|}{\|x\|}}\leq \kappa (A){\frac {\|Ax_{0}-Ax\|}{\|Ax\|}}=\kappa (A){\frac {\|Ax_{0}-b\|}{\|b\|}}~}$

Residual of an approximation to a function

The term residual is used analogously for differential , integral and functional equations , in which instead of a number x, a function is sought that is an equation ${\displaystyle f}$

${\displaystyle T(f)=g}$

Fulfills. For an approximation to , the residual is the function ${\displaystyle f_{0}}$ ${\displaystyle f}$

${\displaystyle g-T(f_{0}).}$

The maximum of the norm of the difference can then be used as a measure of the quality of the approximation

${\displaystyle \max _{x\in {\mathcal {X}}}|g(x)-T(f_{0})(x)|}$

over the range in which the function should approximate the solution or an integral such as ${\displaystyle {\mathcal {X}}}$ ${\displaystyle f_{0}}$ ${\displaystyle f}$

${\displaystyle \int _{\mathcal {X}}|g(x)-T(f_{0})(x)|^{2}~{\rm {d}}x}$

to get voted.

literature

• CT Kelley: Iterative Methods for Linear and Nonlinear Equations. SIAM, ISBN 0-89871-352-8 .
• R. Schaback, H. Wendland: Numerical Mathematics. 5th edition, Springer, 2005.