In this talk I will present our collaborative work on new algorithms for solving two different types of eigenvalue problems. Firstly, a novel orthogonalization-free method together with two specific algorithms are proposed to solve extreme eigenvalue problems. These algorithms achieve eigenvectors instead of eigenspace. Global convergence and local linear convergence are discussed. Efficiency of new algorithms are demonstrated on random matrices and matrices from computational chemistry. Secondly, we explore the possibility of using a reinforcement learning (RL) algorithm to solve large-scale k-sparse eigenvalue problems. By describing how to represent states, actions, rewards and policies, an RL algorithm is designed and demonstrated the effectiveness on examples from quantum many-body physics.
