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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.
We want to solve the following system:
It happens sometimes that the spectral condition number is too high (eigenvalues are not well distributed). Preconditionnement
consists in introducing regular matrix and solving the system:
Let be an intial vector, Preconditioned Gradient Method algorithm
is the following one:
Jacobi Preconditioner consists in taking the diagonal of for the matrix , i.e.
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
We decompose the symmetric matrix like follows: