*Article* **Weighted Block Golub-Kahan-Lanczos Algorithms for Linear Response Eigenvalue Problem**

**Hongxiu Zhong 1,\*, Zhongming Teng 2 and Guoliang Chen 3**


Received: 30 November 2018; Accepted: 29 December 2018; Published: 7 January 2019

**Abstract:** In order to solve all or some eigenvalues lied in a cluster, we propose a weighted block Golub-Kahan-Lanczos algorithm for the linear response eigenvalue problem. Error bounds of the approximations to an eigenvalue cluster, as well as their corresponding eigenspace, are established and show the advantages. A practical thick-restart strategy is applied to the block algorithm to eliminate the increasing computational and memory costs, and the numerical instability. Numerical examples illustrate the effectiveness of our new algorithms.

**Keywords:** linear response eigenvalue problem; block methods; weighted Golub-Kahan-Lanczos algorithm; convergence analysis; thick restart

**AMS Subject Classification:** 65F15; 15A18
