Iterative refinement of the solution of a positive definite system of equations

1966 ◽  
Vol 8 (3) ◽  
pp. 203-216 ◽  
Author(s):  
R. S. Martin ◽  
G. Peters ◽  
J. H. Wilkinson
Keyword(s):  
Positive Definite ◽  
Iterative Refinement ◽  
System Of Equations ◽  
Positive Definite System

scholarly journals Second-refinement of Gauss-Seidel iterative method for solving linear system of equations

2020 ◽  
Vol 13 (1) ◽  
pp. 1-15
Author(s):  
Tesfaye Kebede Enyew ◽  
Gurju Awgichew ◽  
Eshetu Haile ◽  
Gashaye Dessalew Abie
Keyword(s):  
Linear System ◽  
Positive Definite ◽  
System Of Equations ◽  
Linear System Of Equations ◽  
Symmetric Positive Definite ◽  
Modified Method ◽  
Seidel Method ◽  
Diagonally Dominant Matrix ◽  
Diagonally Dominant ◽  
Number Of Iterations

Although large and sparse linear systems can be solved using iterative methods, its number of iterations is relatively large. In this case, we need to modify the existing methods in order to get approximate solutions in a small number of iterations. In this paper, the modified method called second-refinement of Gauss-Seidel method for solving linear system of equations is proposed. The main aim of this study was to minimize the number of iterations, spectral radius and to increase rate of convergence. The method can also be used to solve differential equations where the problem is transformed to system of linear equations with coefficient matrices that are strictly diagonally dominant matrices, symmetric positive definite matrices or M-matrices by using finite difference method. As we have seen in theorem 1and we assured that, if A is strictly diagonally dominant matrix, then the modified method converges to the exact solution. Similarly, in theorem 2 and 3 we proved that, if the coefficient matrices are symmetric positive definite or M-matrices, then the modified method converges. And moreover in theorem 4 we observed that, the convergence of second-refinement of Gauss-Seidel method is faster than Gauss-Seidel and refinement of Gauss-Seidel methods. As indicated in the examples, we demonstrated the efficiency of second-refinement of Gauss-Seidel method better than Gauss-Seidel and refinement of Gauss-Seidel methods.

scholarly journals A parallel Cholesky algorithm for the solution of symmetric linear systems

2004 ◽  
Vol 2004 (25) ◽  
pp. 1315-1327
Author(s):  
R. R. Khazal ◽  
M. M. Chawla
Keyword(s):  
Parallel Algorithm ◽  
Linear Systems ◽  
Triangular Matrix ◽  
Coefficient Matrix ◽  
Positive Definite ◽  
Identity Matrix ◽  
Lower Triangular Matrix ◽  
Elimination Method ◽  
Lower Triangular ◽  
Positive Definite System

For the solution of symmetric linear systems, the classical Cholesky method has proved to be difficult to parallelize. In the present paper, we first describe an elimination variant of Cholesky method to produce a lower triangular matrix which reduces the coefficient matrix of the system to an identity matrix. Then, this elimination method is combined with the partitioning method to obtain a parallel Cholesky algorithm. The total serial arithmetical operations count for the parallel algorithm is of the same order as that for the serial Cholesky method. The present parallel algorithm could thus perform withefficiencyclose to 1 if implemented on a multiprocessor machine. We also discuss theexistenceof the parallel algorithm; it is shown that for a symmetric and positive definite system, the presented parallel Cholesky algorithm is well defined and will run to completion.

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