Geometric interpretation of some Cauchy related methods
For solving systems of linear equations \(Qz = b\) with a symmetric and positive definite matrix \(Q\), a class of iterative methods of the type \(x_{n+1} = x_n + 2 \alpha_n r_n + \alpha_n^2 Q r_n\) with \(r_n = Qx_n - b\) is considered. The real number \(\alpha_n\) is computed from \(x_0, x_1, \dots x_n\), \(Q\), and \(b\). This class includes the Cauchy-Barzilai-Borwein (CBB) method, presented by \textit{M.~Raydan} and \textit{B.~F.~Svaiter} [Comput. Optim. Appl. 21, No. 2, 155--167 (2002; Zbl 0988.90049)] which combines the Cauchy method (steepest descent method) and the Barzilai-Borwein method [see \textit{J.~Barzilai} and \textit{J.~M.~Borwein}, IMA J. Numer. Anal. 8, No. 1, 141--148 (1988; Zbl 0638.65055)]. A geometric interpretation of this class of iterative methods with the help of Bézier parabolas is given. This is used to localize the iterates of the Cauchy, the CBB, and the RBB methods in the plane of the Bézier parabola, where RBB is a combination of the classical Richardson and the Barzilai-Borwein methods. This localization gives some insight into the baviour of these methods. Finally, numerical examples are presented to compare the behaviour of the different methods.
geometric interpretation of iterative methods
scientific article; zbMATH DE number 5977579
scientific article