06027501

j

06027501 Brezinski, C. Novati, P. Redivo-Zaglia, M. A rational Arnoldi approach for ill-conditioned linear systems. J. Comput. Appl. Math. 236, No. 8, 2063-2077 (2012). 2012 Elsevier Science B.V. (North-Holland), Amsterdam EN ill-conditioned linear systems matrix function rational Arnoldi method Tikhonov regularization error estimates condition numbers numerical examples convergence

doi:10.1016/j.cam.2011.09.032

A rational Arnoldi method for the solution of ill-posed full rank linear systems $Ax=b$ is derived by reformulating the problem as $x = f_\lambda(Z_\lambda)b$, $f_\lambda(z) = (1/z - \lambda)^ {-1}$, $Z_\lambda = (A+\lambda I)^{-1}$ and applying the involved matrix-function $f_\lambda$ in a suitable way to the Hessenberg-matrices obtained in the Arnoldi-process. A-priori and a-posteriori error representations are proven. Increasing the parameter $\lambda$ improves the accuracy of the application of $Z_\lambda$ (which means solution of a linear system with matrix $A+\lambda I$) in each Arnoldi-step while at the same time the speed of convergence of the Arnoldi-process decreases. By balancing the condition numbers of $f_\lambda$ and $Z_\lambda$, a strategy for the choice of $\lambda$ is obtained. The performance of the method for unperturbed systems is demonstrated by several numerical examples. Finally, the method is extended to the case of noisy right hand sides $\tilde{b}$ by adapting it to the Tikhonov system $(A^TA + \lambda H^T H)x_\lambda = A^T \tilde{b}$. Martin Rei\ss el (Aachen)