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I will study here the Conjugate Gradient Method. This numerical method allows you to solve linear systems whose matrix is symmetric and positive definite. The search for successive directions makes possible to reach the exact solution of the linear system.
We want to solve the following linear system:
where is a symmetric and positive definite matrix ( and , for all non-zero).
Let be the exact solution of this linear system.
As the matrix is symmetric and positive definite, we can define the following scalar product on :
Two elements are -conjuguate if:
Conjugate Gradient Method consists in building a vectorial sequence of -conjugate vectors . Consequently, the sequence form a basis of . The exact solution can be expanded like follows:
The exact solution is also the unique one minimizer of the functionnal
We have clearly
We define the residual vector of the linear system
is the direction of the gradient of the functional in .
The new direction of descent is the same as its -conjugaison with , we have then:
It is the choice of the coefficient wich allows the -conjugaison of the directions . If you want you can calculate , it is zero !
We calculate the residual for any vector . We fix .