Preconditioned conjugate gradient method

A symmetric preconditioner transforms $Ax=b$ intoThe aim is to make the transformed positive-definite matrix have clustered eigenvalues while keeping application of $P$ inexpensive.

For a positive-definite matrix $A$, choosing $P=A^{-1/2}$ gives $PA{P}^T=I$, so conjugate gradients terminates in one step. Constructing or applying the exact inverse square root is generally as expensive as solving the original system.