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Homotopy understanding of iterative methods. (English) Zbl 1097.65511

The author shows that solving a linear system \(Ax=b\) by iterative methods, such as the Jacobi, Gauss-Seidel, and successive over-relaxation (SOR) methods, can be understood in the setting of homotopy continuation methods, which may provide a new insight into those iterative methods. Based on this new understanding, a condition for the convergence of those iterative methods is presented, which turns out to be equivalent to the traditional one.

MSC:

65F10 Iterative numerical methods for linear systems
65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
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