IV.—Studies in Practical Mathematics. V. On the Iterative Solution of a System of Linear Equations

1950 ◽  
Vol 63 (1) ◽  
pp. 52-60 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Linear Equations ◽  
Iterative Solution ◽  
Positive Definite ◽  
System Of Linear Equations ◽  
Characteristic Roots ◽  
New Variant ◽  
Definite Case ◽  
Practical Mathematics

SynopsisThe convergence of customary processes of iteration for solving linear equations, in particular simple and Seidelian iteration, is studied from the standpoint of matrices. A new variant of Seidelian iteration is introduced. In the positive definite case it always converges, the characteristic roots of its operator being real and positive and less than unity.

Introducing new global electromagnetic modeling solver

2020 ◽  
Author(s):  
Mikhail Kruglyakov ◽  
Alexey Kuvshinov
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Numerical Solution ◽  
Linear Equations ◽  
Numerical Algorithms ◽  
Iterative Solution ◽  
Parallel Computations ◽  
Electromagnetic Modeling ◽  
System Of Linear Equations

<p> In this contribution, we present novel global 3-D electromagnetic forward solver based on a numerical solution of integral equation (IE) with contracting kernel. Compared to widely used x3dg code which is also based on IE approach, new solver exploits alternative (more efficient and accurate) numerical algorithms to calculate Green’s tensors, as well as an alternative (Galerkin) method to construct the system of linear equations (SLE). The latter provides guaranteed convergence of the iterative solution of SLE. The solver outperforms x3dg in terms of accuracy, and, in contrast to (sequential) x3dg, it allows for efficient parallel computations, meaning that the code has practically linear scalability up to the hundreds of processors.</p>

scholarly journals A New Version of the Accelerated Overrelaxation Iterative Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Shi-Liang Wu ◽  
Yu-Jun Liu
Keyword(s):  
Iterative Method ◽  
Linear Equations ◽  
Positive Definite ◽  
System Of Linear Equations ◽  
Positive Definite Matrices ◽  
Symmetric Positive Definite ◽  
Convergence Results ◽  
Aor Method ◽  
Symmetric Positive Definite Matrices ◽  
Overrelaxation Iterative Method

Hadjidimos (1978) proposed a classical accelerated overrelaxation (AOR) iterative method to solve the system of linear equations, and discussed its convergence under the conditions that the coefficient matrices are irreducible diagonal dominant,L-matrices, and consistently orders matrices. In this paper, a new version of the AOR method is presented. Some convergence results are derived when the coefficient matrices are irreducible diagonal dominant,H-matrices, symmetric positive definite matrices, andL-matrices. A relational graph for the new AOR method and the original AOR method is presented. Finally, a numerical example is presented to illustrate the efficiency of the proposed method.

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