Conjugate gradient methods
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For the system of equations: <br> | For the system of equations: <br> | ||
| - | :<math> | + | :<math> A \cdot X = B </math> <br> |
The unpreconditioned conjugate gradient method constructs the '''i'''th iterate <math>x^{(k)}</math> as an element of <math> x^{(k)} + span\left\{ {r^{(0)} ,...,A^{i - 1} r^{(0)} } \right\} </math> so that so that <math> \left( {x^{(0)} - \hat x} \right)^T A\left( {x^{(i)} - \hat x} \right) </math> is minimized , where <math> {\hat x} </math> is the exact solution of <math> AX = B </math>. <br> | The unpreconditioned conjugate gradient method constructs the '''i'''th iterate <math>x^{(k)}</math> as an element of <math> x^{(k)} + span\left\{ {r^{(0)} ,...,A^{i - 1} r^{(0)} } \right\} </math> so that so that <math> \left( {x^{(0)} - \hat x} \right)^T A\left( {x^{(i)} - \hat x} \right) </math> is minimized , where <math> {\hat x} </math> is the exact solution of <math> AX = B </math>. <br> | ||
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. | ||
| + | |||
| + | ==External links== | ||
| + | * [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. | ||
| + | ---- | ||
| + | <i> Return to [[Numerical methods | Numerical Methods]] </i> | ||
Latest revision as of 17:49, 26 August 2006
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.
External links
- Conjugate Gradient Method by N. Soualem.
- Preconditioned Conjugate Gradient Method by N. Soualem.
Return to Numerical Methods
