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# Condition Number

The ratio of the largest to smallest singular value in the singular value decomposition of a matrix. The base- logarithm of is an estimate of how many base- digits are lost in solving a linear system with that matrix. In other words, it estimates worst-case loss of precision. A system is said to be singular if the condition number is infinite, and ill-conditioned if it is too large, where "too large" means roughly the precision of matrix entries.

An estimate of the -norm condition number of a matrix can be computed in the Wolfram Language prior to Version 11.2 using LinearAlgebra`MatrixConditionNumber[m, p] for , 2, or , where omitting the is equivalent to specifying Infinity. A similar approximation for the condition number can be computed using LUDecomposition[mat][[-1]].

## See also

Ill-Conditioned Matrix, Singular Matrix, Singular Value, Singular Value Decomposition

Portions of this entry contributed by Daniel Lichtblau

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## Cite this as:

Lichtblau, Daniel and Weisstein, Eric W. "Condition Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConditionNumber.html