Conjugate gradient methods
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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. | 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. | ||
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| + | <i> Return to [[Numerical methods | Numerical Methods]] </i> | ||
Revision as of 06:25, 3 October 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.
Return to Numerical Methods
