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On a homotopy based method for solving systems of linear equations. (English) Zbl 1331.65061

Summary: A new iterative method is proposed to solve systems of linear algebraic equations, \(Ax = b\). The method is based on the concept of homotopy. Similar works have been done in this direction and some special cases of convergence have been addressed. After reviewing the literature, here we study the method in a more general framework and present more cases of convergence. A comparative study of the method from computational cost viewpoint and speed of convergence shows that the new presented method competes well with classic iterative techniques. Also using a convergence control parameter (CCP) of S. J. Liao [Commun. Nonlinear Sci. Numer. Simul. 14, No. 4, 983–997 (2009; Zbl 1221.65126)] an stationary modification of the method is presented. This modification of the method clarifies two issues, one that Liao’s CCP may fail to be efficient in a linear equation. Also there are cases where this control parameter can extend the convergence cases of the presented homotopy method.

MSC:

65F10 Iterative numerical methods for linear systems
15A06 Linear equations (linear algebraic aspects)
93C05 Linear systems in control theory
65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations

Citations:

Zbl 1221.65126
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