Abstract
The paper presents a dynamic and feasible approach to the successive over-relaxation (SOR) method for solving large scale linear system through iteration. Based on the maximal orthogonal projection technique, the optimal relaxation parameter is obtained by minimizing a derived merit function in terms of right-hand side vector, the coefficient matrix and the previous step values of unknown variables. At each iterative step, we can quickly determine the optimal relaxation value in a preferred interval. When the theoretical optimal value is hard to be achieved, the new method provides an alternative choice of the relaxation parameter at each iteration. Numerical examples confirm that the dynamic optimal successive over-relaxation (DOSOR) method outperforms the classical SOR method.
Highlights
In the paper, we derive a better realization of the successive over-relaxation (SOR) method [Quarteroni, Sacco & Saleri (2000)] to find unknown variables x ∈ Rn from the linear equations system: Ax = b, (1)where A ∈ Rn×n is a given non-singular coefficient matrix and b ∈ Rn is a given right-hand side vector.To minimize the following merit function: { min x f0 =∥b∥2∥Ax∥2 [b · (Ax)]2 } (2)Liu (2013a, 2013b, 2013c, 2014a) has developed new methods to iteratively solve Equation (1)
Numerical examples confirm that the dynamic optimal successive over-relaxation (DOSOR) method outperforms the classical SOR method
2 sin(π∆x) which is inserted into the SOR to find the solution. It converges through 202 iterations, faster than the DOSOR; the maximum error obtained by the SOR with the above wopt is 9.27 × 10−5 and the RMSE is 6.74 × 10−5, which are less accurate than that obtained by the DOSOR with ME=2.11×10−5 and RMSE=1.4×10−5
Summary
Liu (2013a, 2013b, 2013c, 2014a) has developed new methods to iteratively solve Equation (1). The SOR is a well-known and well-developed classical iterative method, and the formulation depends on a relaxation parameter, whose optimal value needs to compute the spectral radius, defined as the absolute value of the largest eigenvalue in magnitude of the iteration matrix. On the other hand, Miyatake, Sogabe & Zhang (2020) proposed an adaptive method based on the Wolfe condition to search the optimal relaxation parameter. Dx + wDx − wUx − wLx = wb + Dx. By using an iterative method to solve the linear system, we suppose that the value xk at the kth-step is known. Where ρ is the spectral radius of In − D−1A
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