Conjugate gradient methods
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Revision as of 22:57, 17 September 2005
Basic Concept
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.
