In numerical analysis, the condition number associated with a problem is a measure of that problem's amenability to digital computation, that is, how numerically well-posed the problem is. A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned.

The condition number of a matrix

For example, the condition number associated with the linear equation $Ax\; =\; b$ gives a bound on how inaccurate the solution $x$ will be after approximate solution. Note that this is before the effects of round-off error are taken into account; conditioning is a property of the matrix, not the algorithm or floating point accuracy of the computer used to solve the corresponding system.

The condition number effectively amplifies the error present in $b$. The extent of this amplification can render a low condition number system (normally a good thing) inaccurate and a high condition number system (normally a bad thing) accurate, depending on how well the data in $b$ are known. For this problem, the condition number is defined by

$\backslash Vert\; A^\{-1\}\backslash Vert\; \backslash cdot\; \backslash Vert\; A\backslash Vert$,

in any consistent norm. This number arises so often in numerical linear algebra that it is given a name, the condition number of a matrix:

$\backslash kappa\; (A)\; =\; \backslash Vert\; A^\{-1\}\backslash Vert\; \backslash cdot\; \backslash Vert\; A\backslash Vert.$

Of course, this definition depends on the choice of norm.

• If $\backslash |\backslash cdot\backslash |$ is $l\_2$ norm then

$\backslash kappa(A)\; =\; \backslash frac\{\backslash sigma\_\{max\}(A)\}\{\backslash sigma\_\{min\}(A)\}$ where $\backslash sigma\_\{max\}(A)$ and $\backslash sigma\_\{min\}(A)$ are maximal and minimal singular values of $A$ respectively. Hence

• If $A$ is normal then

$\backslash kappa(A)\; =\; \backslash left|\backslash frac\{\backslash lambda\_\{max\}(A)\}\{\backslash lambda\_\{min\}(A)\}\backslash right|$ ($\backslash lambda\_\{max\}(A),\backslash \; \backslash lambda\_\{min\}(A)$ are maximal and minimal (by moduli) eigenvalues of $A$ respectively)

• If $A$ is unitary then

$\backslash kappa(A)\; =\; 1$

• If $\backslash |\backslash cdot\backslash |$ is $l\_\{\backslash infty\}$ norm and $A$ is lower triangular non-singular (i.e., $a\_\{ii\}\; \backslash ne\; 0\; \backslash ;\; \backslash forall\; i$) then

$\backslash kappa(A)\; \backslash geq\; \backslash frac\{\backslash max\_i(|a\_\{ii\}|)\}\{\backslash min\_i(|a\_\{ii\}|)\}$

The condition number in other contexts

Condition numbers for singular-value decompositions, polynomial root finding, eigenvalue and many other problems may be defined.

Generally, if a numerical problem is well-posed, it can be expressed as a function $f$ mapping its data, which is an $m$-tuple of real numbers $x$, into its solution, an $n$-tuple of real numbers $y$.

Its condition number is then defined to be the maximum value of the ratio of the relative errors in the solution to the relative error in the data, over the problem domain:

where $\backslash epsilon$ is some reasonably small value in the variation of data for the problem.

If $f$ is also differentiable, this is approximately

$\backslash left|\; \backslash frac\{\; f\text{'}(x)\; \}\{\; f(x)\; \}\; \backslash right|.\; \backslash left|\; x\; \backslash right|$.

Numerical analysis | Kondition (Mathematik) | Число обусловленности | 条件数