# condition number

**condition number** A number that gives a measure of how sensitive the solution of a problem is to changes in the data. In practice such numbers are often difficult to compute; even so they can play an important part in comparing algorithms. They have a particularly important role in numerical linear algebra. As an example, for the linear algebraic equations *A*** x** =

**,**

*b*if

**is changed to**

*b***+ Δ**

*b***(simulating, for example, errors in the data) then the corresponding change Δ**

*b***in the solution satisfies**

*x*where cond(

*A*) = ||

*A*|| ||

*A*

^{–1}|| is the condition number of

*A*with respect to solving linear equations. The expression bounds the relative change in the solution in terms of the relative change in the data

**. The actual quantities are measured in terms of a vector norm (see approximation theory). Similarly the condition number is expressed in terms of a corresponding matrix norm. It can be shown that cond(**

*b**A*)≥1. If cond(

*A*) is large the problem is said to be

*ill-conditioned*and it follows that a small relative change in

**can lead to a large relative change in the solution**

*b***. This means that the accuracy of a computed approximation must be interpreted accordingly, taking into account the size of the possible data errors, machine precision, and errors induced by the particular algorithm.**

*x*Similar ideas apply to other problem areas and condition numbers feature in a measure of eigenvalue sensitivity in the matrix eigenvalue problem.

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**condition number**