Preconditioned Conjugate Gradient Method
Solving linear systems resulting from the finite differences method or of the finite elements shows the limits of the conjugate gradient. Indeed, Spectral condition number of such matrices is too high. The technique of Preconditioned Conjugate Gradient Method consists in introducing a matrix C subsidiary.
Problem
We want to solve the following system:
where
Let
Spectral condition number
It happens sometimes that the spectral condition number
such that the new spectral condition number is smaller for a judicious choice of the matrix
Preconditioned Conjugate Gradient Method
Let
For
EndFor
Jacobi Preconditionner
Jacobi Preconditioner consists in taking the diagonal of
Advantages of such preconditioner are the facility of its implementation and the low memory it needs. But we can find other preconditioners such that resolution of the linear system is fastest, it is the case of the SSOR Preconditioner.
SSOR Preconditioner(Symmetric Successive Over Relaxation)
We decompose the symmetric matrix
where
where
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