**6. Conclusions**

In this paper, we present a weighted block Golub-Kahan-Lanczos algorithm to solve the desired small portion of smallest or largest positive eigenvalues which are in a cluster. Convergence analysis is established in Theorems 1 and 2, and bound the errors of the eigenvalue and eigenvector approximations belonging to an eigenvalue cluster. These results also show the advantages of the block algorithm over the single-vector version. To make the new algorithm more practical, we introduced a thick-restart strategy to eliminate the numerical difficulties caused by the block method. Numerical examples are executed to demonstrate the efficiency of our new restart algorithm.

**Author Contributions:** Conceptualization, G.C.; Data curation, H.Z. and Z.T.; Formal analysis, H.Z. and Z.T.; Methodology, H.Z.; Project administration, H.Z. and G.C.; Resources, H.Z.; Visualization, H.Z. and Z.T.; Writing—original draft, H.Z.; Writing—review and editing, Z.T. and G.C.

**Funding:** This work was financial supported by the National Nature Science Foundation of China (No. 11701225, 11601081, 11471122), Fundamental Research Funds for the Central Universities (No. JUSRP11719), Natural Science Foundation of Jiangsu Province (No. BK20170173), and the research fund for distinguished young scholars of Fujian Agriculture and Forestry University (No. xjq201727).

**Conflicts of Interest:** The authors declare no conflict of interest.
