(Difference between revisions)
 Revision as of 18:39, 19 August 2006 (view source)Nsoualem (Talk | contribs) (→Basic Concept)← Older edit Latest revision as of 17:49, 26 August 2006 (view source)Nsoualem (Talk | contribs) (→External links) Line 9: Line 9: ==External links== ==External links== * [http://www.math-linux.com/spip.php?article54 Conjugate Gradient Method] by N. Soualem. * [http://www.math-linux.com/spip.php?article54 Conjugate Gradient Method] by N. Soualem. + * [http://www.math-linux.com/spip.php?article55 Preconditioned Conjugate Gradient Method] by N. Soualem. ---- ---- Return to [[Numerical methods | Numerical Methods]] Return to [[Numerical methods | Numerical Methods]]

## Basic Concept

For the system of equations:

$A \cdot X = B$

The unpreconditioned conjugate gradient method constructs the ith iterate $x^{(k)}$ as an element of $x^{(k)} + span\left\{ {r^{(0)} ,...,A^{i - 1} r^{(0)} } \right\}$ so that so that $\left( {x^{(0)} - \hat x} \right)^T A\left( {x^{(i)} - \hat x} \right)$ is minimized , where ${\hat x}$ is the exact solution of $AX = B$.

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.