Abstract
In this chapter we consider general properties of iterative methods. Such properties are consistency, ensuring the connection between the iterative method and the given system of equations, as well as convergence, guaranteeing the success of the iteration. The most important result of this chapter is the characterisation of the convergence of linear iterations by the spectral radius of the iteration matrix (cf. §2.1.4). Since we only consider iterative methods for systems with regular matrices, iterative methods for singular systems or those with rectangular matrices will not be studied (Concerning this topic, we refer, e.g., to Björck [47], Marek [275], Kosmol–Zhou [241], Berman–Plemmons [46], and Remark 5.17). The quality of a linear iteration depends on both the cost and the convergence speed. The resulting efficacy is discussed in Section 2.3. Finally, Section 2.4 explains how to test iterative methods numerically.
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Hackbusch, W. (2016). Iterative Methods. In: Iterative Solution of Large Sparse Systems of Equations. Applied Mathematical Sciences, vol 95 . Springer, Cham. https://doi.org/10.1007/978-3-319-28483-5_2
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DOI: https://doi.org/10.1007/978-3-319-28483-5_2
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-28481-1
Online ISBN: 978-3-319-28483-5
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