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Choosing the relaxation parameter in sequential block-iterative methods for linear systems

In this paper we introduce two strategies for picking relaxation parameters to control the semiconvergence behavior of a sequential block-iterative method. A convergence analysis is presented. We also demonstrate the performance of our strategies by examples taken from tomographic imaging. | Turk J Math (2017) 41: 733 – 748 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1503-6 Research Article Choosing the relaxation parameter in sequential block-iterative methods for linear systems Touraj NIKAZAD∗, Shaghayegh HEIDARZADE School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran Received: 02.03.2015 • Accepted/Published Online: 20.07.2016 • Final Version: 22.05.2017 Abstract: In this paper we introduce two strategies for picking relaxation parameters to control the semiconvergence behavior of a sequential block-iterative method. A convergence analysis is presented. We also demonstrate the performance of our strategies by examples taken from tomographic imaging. Key words: Sequential block-iterative methods, Cimmino and CAV iteration, semiconvergence, relaxation parameters, tomographic imaging 1. Introduction Ill-posed and large-scale problems, such as computed tomography, take place in many fields of mathematics and physical sciences. Usually these problems are handled by iterative methods instead of direct methods. There is an interest in regularizing iterative methods where the iteration vector can be considered as a regularized solution. Using incorrect and noisy input data, which are due to measurements or rounding errors, we obtain a more difficult problem to solve. The iteration index of an iterative method may be considered as a regularization parameter. Initially the iteration vectors approach a regularized solution. Nevertheless, continuing the iteration process often produces iteration vectors that are corrupted by noise; see [16, p. A2002] and [19, p. 1]. This phenomenon was called semiconvergence by Natterer [30, page 157]; for analysis of the phenomenon, see, e.g., [4, 20, 21, 23, 33, 35]. The typical overall error behavior is shown in Figure 1. If there is a reliable stopping rule then we may get a proper approximation of the sought solution, i.e. x∗