Volume 47, pp. 153-178, 2017.

Weighted Golub-Kahan-Lanczos bidiagonalization algorithms

Hong-Xiu Zhong and Hongguo Xu

Abstract

We present weighted Golub-Kahan-Lanczos algorithms. We demonstrate their applications to the eigenvalue problem of a product of two symmetric positive definite matrices and an eigenvalue problem for the linear response problem. A convergence analysis is provided and numerical test results are reported. As another application we make a connection between the proposed algorithms and the preconditioned conjugate gradient (PCG) method.

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Key words

weighted Golub-Kahan-Lanczos bidiagonalization algorithm, eigenvalue, eigenvector, Ritz value, Ritz vector, linear response eigenvalue problem, Krylov subspace, bidiagonal matrices

AMS subject classifications

65F15, 15A18

ETNA articles which cite this article

Vol. 51 (2019), pp. 529-546 Hong-Xiu Zhong and Guo-Liang Chen: Thick restarting the weighted harmonic Golub-Kahan-Lanczos algorithm for the linear response eigenvalue problem