## Quick Reference

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 *b* is changed to *b* + Δ*b* (simulating, for example, errors in the data) then the corresponding change Δ*x* in the solution satisfieswhere 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 *b*. 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(*A*)≥1. If cond(*A*) is large the problem is said to be **ill-conditioned** and it follows that a small relative change in *b* can lead to a large relative change in the solution *x*. This means that the accuracy of a computed approximation must be interpreted accordingly, taking into account the size of the isible data errors, machine precision, and errors induced by the particular algorithm.

*A**x* = *b*,

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

*Subjects:*
Computing.

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