Preconditioner

Summary

A preconditioner for a linear system $A x = b$ is an invertible matrix $P$ that is close to $A$ in a suitable sense. Systems $P z = r$ must be cheap to solve. One replaces the system by $P^{-1} A x = P^{-1} b$, or by a similar transformed system. The new matrix has a smaller condition number or better clustered eigenvalues. Iterative methods then converge in far fewer steps. For large sparse systems, the choice of preconditioner usually matters more than the choice of iterative method.