An efficient method, preconditioned conjugate gradient method with a filtering function (PCG-F), is proposed for solving iteratively the Dirac equation in 3D lattice space for nuclear systems by defining a variational subspace, in which the positive-energy solutions are minima rather than saddle points as in the full variational space. This allows one to obtain the single nucleon energies and wavefunctions of the Dirac equation in 3D lattice space efficiently. The PCG-F method is demonstrated in solving the Dirac equation with given spherical and deformed Woods-Saxon potentials. The solutions given by the inverse Hamiltonian method in 3D lattice space and the shooting method in radial coordinate space are reproduced with a high accuracy. In comparison with the existing inverse Hamiltonian method, the present PCG-F method is much faster in the convergence of the iteration, in particular for deformed potentials. It may also provide a promising way to solve the relativistic Hartree-Bogoliubov equation iteratively in the future.

中文翻译：

---

共轭梯度法在3D晶格空间中Dirac方程的有效解

提出了一种有效的方法，即具有滤波功能的预处理共轭梯度法（PCG-F），通过定义变分子空间来迭代求解核系统的3D晶格空间中的狄拉克方程，在该子空间中，正能量解为最小值而不是鞍点位于整个变分空间中。这使得人们可以有效地获得3D晶格空间中Dirac方程的单核子能量和波函数。在求解具有给定的球形和变形Woods-Saxon势的Dirac方程时，证明了PCG-F方法。可以高精度地再现3D晶格空间中的汉密尔顿逆方法和径向坐标空间中的射击方法。与现有的哈密顿逆方法相比，当前的PCG-F方法在迭代收敛方面要快得多，尤其是对于变形的电势而言。它也可能为将来迭代求解相对论性Hartree-Bogoliubov方程提供一种有希望的方法。