Abstract
In this paper, we propose another version of the full orthogonalization method (FOM) for the resolution of linear system Ax = b, based on an extended definition of Sturm sequence in the calculation of the determinant of an upper hessenberg matrix in o(n2). We will also give a new version of Givens method based on using a tensor product and matrix addition. This version can be used in parallel calculation.
Highlights
The resolution of linear systems Ax = b is in the heart of numerous scientific applications: discretization of partiel differentiel equations, image processing, the linearization of nonlinear problems in a sequence of linear problems [1] [2] [3] etc
Most iterative methods [2] [6] [7] treat linear systems through vector-matrix products with an appropriate data structure permitting the exploitation of sparsity of the matrix A by storing only nonzero elements which are indispensable for the calculation of these products, which reduces both the memory size and the processing time
The most used Krylov methods based on Arnoldi algorithm are: the full orthogonalization method (FOM) and its varieties: the method of minimal residue, Lanczos method, and that of conjugate gradient for symmetric and symmetric positive definite matrices
Summary
Among the most known ones, we mention Jacobi’s, Gauss-Seidel’s and the relaxation methods These methods remain less reliable than the direct methods, and more or less efficient with some specific problems. The most used Krylov methods based on Arnoldi algorithm are: the full orthogonalization method (FOM) and its varieties: the method of minimal residue (or shortly GMRES), Lanczos method, and that of conjugate gradient for symmetric and symmetric positive definite matrices. These projection methods are more generalisable, more powerful and currently the most used.
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