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One of the most efficient methods for solving the polynomial eigenvalue problem (PEP) is the Sakurai-Sugiura method with Rayleigh-Ritz projection (SS-RR), which finds the eigenvalues contained in a certain domain using the contour integral. The SS-RR method converts the original PEP to a small projected PEP using the Rayleigh-Ritz projection. However, the SS-RR method suffers from backward instability when the norms of the coefficient matrices of the projected PEP vary widely. To improve the backward stability of the SS-RR method, we combine it with a balancing technique for solving a small projected PEP. We then analyze the backward stability of the SS-RR method. Several numerical examples demonstrate that the SS-RR method with the balancing technique reduces the backward error of eigenpairs of PEP.  相似文献
2.
We consider solving complex symmetric linear systems with multiple right-hand sides. We assume that the coefficient matrix has indefinite real part and positive definite imaginary part. We propose a new block conjugate gradient type method based on the Schur complement of a certain 2-by-2 real block form. The algorithm of the proposed method consists of building blocks that involve only real arithmetic with real symmetric matrices of the original size. We also present the convergence property of the proposed method and an efficient algorithmic implementation. In numerical experiments, we compare our method to a complex-valued direct solver, and a preconditioned and nonpreconditioned block Krylov method that uses complex arithmetic.  相似文献
3.
We investigate contour integral-based eigensolvers for computing all eigenvalues located in a certain region and their corresponding eigenvectors. In this paper, we focus on a Rayleigh–Ritz type method and analyze its error bounds. From the results of our analysis, we conclude that the Rayleigh–Ritz type contour integral-based eigensolver with sufficient subspace size can achieve high accuracy for target eigenpairs even if some eigenvalues exist outside but near the region.  相似文献
4.
We investigate the restart of the Restarted Shifted GMRES method for solving shifted linear systems.Recently the variant of the GMRES(m) method with the unfixed update has been proposed to improve the convergence of the GMRES(m) method for solving linear systems,and shown to have an efficient convergence property.In this paper,by applying the unfixed update to the Restarted Shifted GMRES method,we propose a variant of the Restarted Shifted GMRES method.We show a potentiality for efficient convergence within the variant by some numerical results.  相似文献
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