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Additive preconditioning, eigenspaces, and the inverse iteration. (English) Zbl 1159.65043
In modern eigensolvers such as the inverse power method, the Jacobi-Davidson algorithm, and the Arnoldi iteration with shift-and-invert technique the iteration requires the solution of a system of linear algebraic equations for every loop. Because the systems are prevalently ill-conditioned in general factorizations of the input matrix have to be used to overcome accuracy and iteration number problems in the inverse iteration. The ill-conditioning of the matrix is especially a disadvantage in the case of large-scale problems that need the application of iterative algorithms for the solution of the linear systems.
The main feature of the paper consists in the application of an additive randomized preconditioning technique, already developed in the previous paper by V. Y. Pan, D. Ivolgin, B. Morphy, R. E. Rosholt, Y. Tang and X. Yan [Computer science – theory and applications. Third international computer science symposium in Russia, CSR 2008 Moscow, Russia, June 7–12, 2008. Proceedings. Berlin: Springer. Lecture Notes in Computer Science 5010, 372–383 (2008; Zbl 1142.68607)] to the systems to improve the conditioning of the input matrix and subsequently to reduce the iteration number and to increase the accuracy in the inverse iteration covering also the cases of multiple and clustered eigenvalues. It is proved that the quadratic convergence of the inverse method is preserved. The fast global convergence is claimed by experiments. The results are verified by numerical examples of small matrix order ($$n = 64$$, $$n = 100$$).

##### MSC:
 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 65F35 Numerical computation of matrix norms, conditioning, scaling
##### Software:
mctoolbox; Eigensolve; JDQR; JDQZ
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