An a posteriori verification method is proposed for the generalized real-symmetric eigenvalue problem and is applied to densely clustered eigenvalue problems in large-scale electronic state calculations. The proposed method is realized by a two-stage process in which the approximate solution is computed by existing numerical libraries and is then verified in a moderate computational time. The procedure returns intervals containing one exact eigenvalue in each interval. Test calculations were carried out for organic device materials, and the verification method confirms that all exact eigenvalues are well separated in the obtained intervals. This verification method will be integrated into EigenKernel (https://github.com/eigenkernel/), which is middleware for various parallel solvers for the generalized eigenvalue problem. Such an a posteriori verification method will be important in future computational science.

针对广义实对称特征值问题，提出了一种后验验证方法，并将其应用于大规模电子状态计算中的密集簇特征值问题。所提出的方法是通过两步过程实现的，该过程中的近似解由现有的数值库计算，然后在适度的计算时间内得到验证。该过程返回在每个间隔中包含一个准确特征值的间隔。对有机器件材料进行了测试计算，验证方法确认了在获得的间隔中所有精确的特征值均已很好地分离。该验证方法将集成到EigenKernel（https://github.com/eigenkernel/）中，该工具是用于广义特征值问题的各种并行求解器的中间件。