Conjugate gradient methods
For the system of equations:
The unpreconditioned conjugate gradient method constructs the ith iterate as an element of so that so that is minimized , where is the exact solution of .
This minimum is guaranteed to exist in general only if A is symmetric positive definite. The preconditioned version of these methods use a different subspace for constructing the iterates, but it satisfies the same minimization property over different subspace. It requires that the preconditioner M is symmetric and positive definite.