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# Condition Number Definition/Meaning:

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

Ax = b,

if b is changed to b + ∆b (simulating, for example, errors in the data) then the corresponding change x in the solution satisfies

||∆x|| / ||x|| ≤ cond (A) ||∆b|| / ||b||

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 b. The actual quantities are measured in terms of a vector norm. Similarly the condition number is expressed in terms of a corresponding matrix norm. It can be shown that cond(A) l. 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 any precision in the solution is spurious.

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