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Linear Simultaneous Algebraic Equations and Methods of Obtaining Their Solutions
Published in Karan S. Surana, Numerical Methods and Methods of Approximation, 2018
In this section we examine the conditions under which the Jacobi method will converge. Consider: () {x}j+1={b^}−([A^]−[I]){x}j
Iterative regularization methods
Published in Mario Bertero, Patrizia Boccacci, Christine De Mol, Introduction to Inverse Problems in Imaging, 2021
Mario Bertero, Patrizia Boccacci, Christine De Mol
A very simple iterative method for approximating the least-squares solutions of first-kind integral equations has been introduced independently by Landweber [189] and Fridman [123]. For this reason, the method is usually called the Landweber or the Landweber-Fridman method, at least in the western literature concerning ill-posed problems. In the Russian literature the denomination of successive-approximations method is preferred. It is probably more correct to call it the Jacobi method because the basic idea is the same as in an iterative method introduced by Jacobi for solving linear algebraic systems.
Review of Basic Laws and Equations
Published in Pradip Majumdar, Computational Methods for Heat and Mass Transfer, 2005
It can be observed in the Jacobi method that all unknown values are computed using values estimated in the previous iteration step, rather than using most recent values of the other unknowns. Considerable improvement in the storage requirements and the rate of convergence are achieved in many problems by employing an alternative iterative scheme known as the Gauss-Seidel method, in which as unknowns are computed in an iteration step; they are subsequently employed in the computation of the rest of the unknown in the same iteration step. This iterative scheme can be expressed as xik+1=ci−∑j=1i−1aijxjk+1−∑j=i+1naijxjkaii,fori=1,2,…,n
Enhanced idealized explicit FEM for predicting welding deformation in complex large-scaled structure and application to the real structure
Published in Welding International, 2019
Kazuki Ikushima, Shintaro Maeda, Teruya Ieshita, Atsushi Kawahara, Yuuta Abe, Hirotaka Kiuchi, Masakazu Shibahara
Furthermore, in conventional multigrid methods, a steady iteration such as the Jacobi method is frequently used in the calculation part as in Figure 3 but idealized explicit FEM was used in this study. This is because since a steady iteration method such as the Gauss-Seidel method or SOR method involves sequential processing, parallelization is difficult. Parallelization can be easily applied to the Jacobi method but since a condition required for convergence with the Jacobi method is that the coefficient matrix is strictly diagonally dominant, it is highly possible that convergence cannot be obtained stably and it is unused for reasons of stability of analysis.