A preconditioner for a linear system 
is an invertible matrix 
for which systems with 
are substantially easier to solve than systems with 
, while 
has more favorable spectral or conditioning properties.
Left preconditioning replaces the system by
 |
and right preconditioning instead solves 
with 
.
Effective preconditioning can greatly reduce the iterations required by a conjugate gradient method or another Krylov-subspace method.
