The Jacobi, Gauss-Seidel and SOR methods belong to the class of simple iterative methods for linear systems. Because of the parameter , the SOR method is more effective than the Gauss-Seidel method. Here, a new approach to the simple iterative methods is proposed. A new parameter q can be introduced to every simple iterative method. Then, if a matrix of a system is positive definite and the parameter q is sufficiently large, the method is convergent. The original Jacobi method is convergent only if the matrix is diagonally dominated, while the Jacobi method with the parameter q is convergent for every positive definite matrix. The optimality criterion for the choice of the parameter q is given, and thus, interesting results for the Jacobi, Richardson and Gauss-Seidel methods are obtained. The Gauss-Seidel method with the parameter q, in a sense, is equivalent to the SOR method. From the formula for the optimal value of q results the formula for optimal value of . Up to present, this formula was known only in special cases. Practical useful approximate formula for optimal value is also given. The influence of the parameter q on the speed of convergence of the simple iterative methods is shown in a numerical example. Numerical experiments confirm: for very large scale systems the speed of convergence of the SOR method with optimal or approximate parameter is near the same (in some cases better) as the speed of convergence of the conjugate gradients method.
Published in | Applied and Computational Mathematics (Volume 8, Issue 5) |
DOI | 10.11648/j.acm.20190805.11 |
Page(s) | 82-87 |
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Iterative Methods for Linear Systems, Optimal Parameter for SOR Method, Positive Definite Matrices
[1] | I. Koutis, G. L. Miller, R. Peng, A fast solver for a class of linear systems, Communications of the ACM 55 (10) (2010) 99-107. |
[2] | E. G. Boman, B. Hendrickson, S. Vavasis, Solving elliptic finite element systems in near-linear time with support preconditioners, SIAM J. Numer. Anal. 46 (6) (2008) 3264-3284. |
[3] | J. D. Hogg, J. A. Scott, A fast and robust mixed-precision solver for the solution of sparse symmetric linear systems, ACM Trans. on Math. Soft. 37 (2) (2010). |
[4] | J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Third Ed., Springer Verlag, 2002. |
[5] | Y. Saad, Iterative Methods for Sparse Linear Systems, Second Ed., SIAM Press, 2016. |
[6] | D. S. Watkins, Fundamentals of Matrix Computations, Second Ed., Wiley, 2002. |
[7] | D. M. Young, Iterative Solutions of Large Linear Systems, Academic Press, New York, 1971, reprinted by Dover 2003. |
[8] | C. G. Broyden, On the convergence criteria for the method of successive over-relaxation, Mathematics of Computation 18 (85) (1964) 136-141. |
[9] | S. Yang, M. K. Gobbert, The optimal relaxation parameter for the SOR method applied to the Poisson equation in any space dimensions, Appl. Math. Letters, 22 (2009) 325-331. |
[10] | J. W. Demmel, Applied Numerical Linear Algebra, SIAM Press, 1997. |
[11] | I. K. Youssef, S. M. Ali, M. Y. Hamada, On the Line Successive Overrelaxation Method, Applied and Computational Mathematics, 5, 3 (2016) 103-106. |
[12] | Cheng-yi Zhang, Zichen Xue, A convergence analysis of SOR iterative methods for linear systems with week H-matrices, Open Mathematics, 2016. |
[13] | S. Karunanithi, N. Gajalakshmi, M. Malarvishi, M. Saileshwari, A Study on comparison of Jacobi, Gauss-Seidel and SOR methods for the solution in system of linear equations, Int. J. of Math. Trends and Technology, (IJMTT), 56, 4, 2018. |
[14] | O. Nevanlinna, How fast can iterative methods be? In “Recent Advances in Iterative Methods”, ed. by G. Golub, A. Greenbaum, M. Luskin, Math. and its Appl., 60 (2012) 135-148. |
[15] | H. Woźniakowski, Round-off error analysis of iterations for large linear systems, Numer. Math., 30 (1978) 301-314. |
APA Style
Stanislaw Marian Grzegorski. (2019). On Optimal Parameter Not Only for the SOR Method. Applied and Computational Mathematics, 8(5), 82-87. https://doi.org/10.11648/j.acm.20190805.11
ACS Style
Stanislaw Marian Grzegorski. On Optimal Parameter Not Only for the SOR Method. Appl. Comput. Math. 2019, 8(5), 82-87. doi: 10.11648/j.acm.20190805.11
AMA Style
Stanislaw Marian Grzegorski. On Optimal Parameter Not Only for the SOR Method. Appl Comput Math. 2019;8(5):82-87. doi: 10.11648/j.acm.20190805.11
@article{10.11648/j.acm.20190805.11, author = {Stanislaw Marian Grzegorski}, title = {On Optimal Parameter Not Only for the SOR Method}, journal = {Applied and Computational Mathematics}, volume = {8}, number = {5}, pages = {82-87}, doi = {10.11648/j.acm.20190805.11}, url = {https://doi.org/10.11648/j.acm.20190805.11}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20190805.11}, abstract = {The Jacobi, Gauss-Seidel and SOR methods belong to the class of simple iterative methods for linear systems. Because of the parameter , the SOR method is more effective than the Gauss-Seidel method. Here, a new approach to the simple iterative methods is proposed. A new parameter q can be introduced to every simple iterative method. Then, if a matrix of a system is positive definite and the parameter q is sufficiently large, the method is convergent. The original Jacobi method is convergent only if the matrix is diagonally dominated, while the Jacobi method with the parameter q is convergent for every positive definite matrix. The optimality criterion for the choice of the parameter q is given, and thus, interesting results for the Jacobi, Richardson and Gauss-Seidel methods are obtained. The Gauss-Seidel method with the parameter q, in a sense, is equivalent to the SOR method. From the formula for the optimal value of q results the formula for optimal value of . Up to present, this formula was known only in special cases. Practical useful approximate formula for optimal value is also given. The influence of the parameter q on the speed of convergence of the simple iterative methods is shown in a numerical example. Numerical experiments confirm: for very large scale systems the speed of convergence of the SOR method with optimal or approximate parameter is near the same (in some cases better) as the speed of convergence of the conjugate gradients method.}, year = {2019} }
TY - JOUR T1 - On Optimal Parameter Not Only for the SOR Method AU - Stanislaw Marian Grzegorski Y1 - 2019/11/22 PY - 2019 N1 - https://doi.org/10.11648/j.acm.20190805.11 DO - 10.11648/j.acm.20190805.11 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 82 EP - 87 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20190805.11 AB - The Jacobi, Gauss-Seidel and SOR methods belong to the class of simple iterative methods for linear systems. Because of the parameter , the SOR method is more effective than the Gauss-Seidel method. Here, a new approach to the simple iterative methods is proposed. A new parameter q can be introduced to every simple iterative method. Then, if a matrix of a system is positive definite and the parameter q is sufficiently large, the method is convergent. The original Jacobi method is convergent only if the matrix is diagonally dominated, while the Jacobi method with the parameter q is convergent for every positive definite matrix. The optimality criterion for the choice of the parameter q is given, and thus, interesting results for the Jacobi, Richardson and Gauss-Seidel methods are obtained. The Gauss-Seidel method with the parameter q, in a sense, is equivalent to the SOR method. From the formula for the optimal value of q results the formula for optimal value of . Up to present, this formula was known only in special cases. Practical useful approximate formula for optimal value is also given. The influence of the parameter q on the speed of convergence of the simple iterative methods is shown in a numerical example. Numerical experiments confirm: for very large scale systems the speed of convergence of the SOR method with optimal or approximate parameter is near the same (in some cases better) as the speed of convergence of the conjugate gradients method. VL - 8 IS - 5 ER -