TY - JOUR
T1 - An a posteriori verification method for generalized real-symmetric eigenvalue problems in large-scale electronic state calculations
AU - Hoshi, Takeo
AU - Ogita, Takeshi
AU - Ozaki, Katsuhisa
AU - Terao, Takeshi
N1 - Funding Information:
The authors wish to thank the anonymous referees for their valuable comments, which helped to improve our paper significantly. The present study was supported in part by MEXT as Exploratory Issue 1–2 of the Post-K (Fugaku) computer project “Development of verified numerical computations and super high-performance computing environment for extreme researches” using computational resources of the K computer provided by the RIKEN R-CCS through the HPCI System Research project, Japan (Project ID: hp180222) and Priority Issue 7 of the Post-K computer project, Japan and by JSPS KAKENHI, Japan Grant Numbers 16KT0016 , 17H02828 , and 19H04125 .
Publisher Copyright:
© 2020 Elsevier B.V.
PY - 2020/10/1
Y1 - 2020/10/1
N2 - 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.
AB - 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.
KW - Electronic state calculation
KW - Generalized real-symmetric eigenvalue problem
KW - Supercomputer
KW - Verification method
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U2 - 10.1016/j.cam.2020.112830
DO - 10.1016/j.cam.2020.112830
M3 - Article
AN - SCOPUS:85081133612
VL - 376
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
SN - 0377-0427
M1 - 112830
ER -