Preconditioner

In linear algebra and numerical analysis, a preconditioner P of a matrix A is a matrix such that P⁻¹A has a smaller condition number than A. A preconditioner as defined here is also known as a left preconditioner; there are also right preconditioners, but they are not discussed in this article.

Preconditioners are useful when using an iterative method to solve a large, sparse linear system

$Ax=b\,$

for x. Solving the preconditioned system

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

is equivalent to solving the original system. However, the preconditioned system has a smaller condition number; it is better conditioned. The number of iterations used to solve the system increases with the condition number so by reducing the number of needed iterations (if the cost — computing time — of applying P^-1 is small), a gain in efficiency is achieved.

The matrix P⁻¹A is never explicitly formed. In using an iterative method, only the action of applying P⁻¹ to a given vector need be computed.

See also

• Krylov subspace
• Iterative methods
• Conjugate gradient method