Abstract
We present several iterations for preconditioners introduced by Tarazaga and Cuellar (2009), and study the convergence of the method for solving a linear system whose coefficient matrix is positive definite matrices, and we also find that they complete very well with the SOR iteration, which is shown through numerical examples.
Highlights
For solving the large sparse linear systemAx = b, x, b ∈ Rn, (1)where A ∈ Rn×n is a square nonsingular positive definite matrix, an iteration method is often considered
We present several iterations for preconditioners introduced by Tarazaga and Cuellar (2009), and study the convergence of the method for solving a linear system whose coefficient matrix is positive definite matrices, and we find that they complete very well with the SOR iteration, which is shown through numerical examples
Xk+1 = (I − ω(I + ωPfL)−1PfA) xk + (I + ωPfL)−1ωPfb. (15). This iteration is considered as sequential Frobenius norm iteration
Summary
Where A ∈ Rn×n is a square nonsingular positive definite matrix, an iteration method is often considered. Where P is called the preconditioner or a preconditioning matrix. A matrix A = (aij) is called a row diagonally dominant if. Because I + ωPfL is inverse, we can get xk+1 = (I − (Pf−1 + ωL)−1A) xk + (Pf−1 + ωL)−1ωb (14) This iteration is considered as sequential Frobenius norm iteration. We obtained and built the infinity norm iteration associated with P∞ as follows: xk+1 = (I − (P∞−1 + ωL)−1A) xk + (P∞−1 + ωL)−1ωb (16). We have set two preconditioned SOR iterative methods which use Pf and P∞ as a preconditioner. First, we discuss the convergence of the preconditioned SOR iterative method which uses Pf and P∞ as a preconditioner.
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