In linear algebra and numerical analysis a preconditioner P of a matrix A is a matrix such that P^-1A has a lower condition number than A.

This is useful, for instance, when solving a linear system

$Ax\; =\; b,\backslash ,$

for x using iterative methods, since the number of iterations usually increases with the condition number. Thus, solving

$P^\{-1\}Ax\; =\; P^\{-1\}b,\backslash ,$

is equivalent to the original system but has a lower condition number, i.e. it is better conditioned. If the number of iterations are consequently reduced and the cost (computing time) of applying P^-1 is small, a gain in total computing time is achieved.

See also

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• Krylov subspace
• Iterative methods
• Conjugate gradient method

Numerical linear algebra