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Conjugate gradient methods

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==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.
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<i> Return to [[Numerical methods | Numerical Methods]] </i>
<i> Return to [[Numerical methods | Numerical Methods]] </i>

Latest revision as of 17:49, 26 August 2006

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.

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