四元数量子力学是现代量子力学的一个重要分支，本项目主要研究四元数量子力学中的两类数学问题：四元数多项式方程 (QPE) 和四元数最小二乘 (QLS) 问题。给出一般的双边二次四元数多项式方程有解的充要条件，提供求解一般双边二次四元数多项式方程的算法；给出求四元数最小二乘问题极小范数解的迭代算法，使较大规模四元数最小二乘问题的计算得以实现，来源于实际物理问题中的数学模型能够快速求解，并用数值实验检验所给算法的有效性。

Quaternionic quantum mechanics is a important branch in modern quntum mechanics, we mainly concern with two classes of mathematic problems in quaternionic quantum mechanics: quaternion polynomial equation (QPE) and quaternion Least Squares (QLS) problem. We will give the necessary and sufficient conditions for the QPE having a solution, and provide algorithms for solving the general couple QPE; in addition, we will derive algorithms for finding the minimum norm solution of the QLS problem, which satisfies with large scaled QLS problems that root in practical physics problems. We will verify the efficiency of our algorithms by numerical experiments.