# Difference between explicit and implicit preconditioning

What is the difference between an explicit and implicit preconditioner?

An explicit preconditioner $G \approx A^{-1}$ approximates the inverse of the system matrix $A$, an implicit preconditioner $C \approx A$ the system itself. Correspondingly, application of the explicit preconditioner $G$ amounts to multiplication with $G$, while application of the implicit preconditioner $C$ amounts to solving a linear system with coefficient matrix $C$.