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.
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