线性响应特征值问题在量子力学、核物理和凝聚态物理学等领域具有广泛的应用，然而，该问题的数值算法仍存在许多待完善的地方。本项目在前期工作的基础上，将深入研究以下几个内容：（1）通过深入挖掘线性响应特征值问题的特殊结构性质，构造投影矩阵为Hermitian矩阵的加权Golub-Kahan-Lanczos算法，减少对问题求解的运算量与存储量；（2）基于调和投影技术，提出调和加权Golub-Kahan-Lanczos算法，并建立算法的Rayleigh-Ritz收敛性理论，这也是应用斜投影方法于线性响应特征值问题的另一个有益探索；（3）推广新算法至Block形式，并考虑其加速技术，能够解决同时求解多个特征值的需要。本项目提出的一整套新型子空间类算法及其理论，有望发展并促进线性响应特征值问题的理论与算法体系的完善，研究结果将为电子结构系统中能量激发态的计算提供高效的计算方法以及相应的理论依据。

Linear response eigenvalue problems are widely used in quantum mechanics, nuclear physics and condensed matter physics. However, the numerical algorithms on these problems still need to be perfect. Based on our previous research, this project aims to study the following problems. (1) Through fully exploiting the special structures of linear response eigenvalue problems, we will design weighted Golub-Kahan-Lanczos algorithm with the projection matrix being Hermitian to reduce the cost and storage. (2) Based on the harmonic projection, the harmonic weighted Golub-Kahan-Lanczos algorithm will be proposed, and its Rayleigh-Ritz convergence theories will also be established. It is a beneficial exploration for linear response eigenvalue problems when using the oblique projection methods. (3) The block form of the above algorithms are extended to solve multiple eigenvalues simultaneously. At the same time, we also consider the accelerating strategy, and the convergence theories. The new theories and new subspaces algorithms proposed in this project aims to develop and promote the theory and algorithms systems of the linear response eigenvalue problems. Therefore, the research results will provide efficient algorithms and theoretical basis for the computations of energy excitation in electronic structure system.