Condition Number

The ratio C of the largest to smallest singular value in the singular value decomposition of a matrix. The base-b logarithm of C is an estimate of how many base-b 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 log(C)>~ the precision of matrix entries.

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

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