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A new Lanczos method for electronic structure calculations

Content Provider	ACM Digital Library
Author	Simon, H. D. Canning, A. Wu, K.
Abstract	In the heart of most electronic structure simulation programs, there is a routine to find the solution of eigenvalue problems. Solving these eigenvalue problems usually dominates the computer time used for the whole simulation[6]. Because of their physical properties, these eigenvalue problems are always symmetric real or Hermitian. The dimensions of the matrices involved are usually very large and a large number of eigenvalues and their corresponding eigenvectors are needed to compute the desired physical quantities. To solve this type of problems, we introduce a variant of the Lanczos method called the thick-restart Lanczos method. In material science, this method is most appropriate for non-selfconsistent cases where the eigenvalue problems are linear and the number of required eigenvalues is relatively small compared to the size of the matrix.The Lanczos method is very simple and yet effective in finding eigenvalues. It is also well suited for parallel computing. There are two common ways of implementing the Lanczos method depending on whether the Lanczos vectors are stored. When the Lanczos vectors are not stored, they may lose orthogonality and the Lanczos method may generate spurious eigenvalues [2, 10]. Though spurious eigenvalues can be effectively identified, we still prefer not to deal with the spurious eigenvalues. When the Lanczos vectors are stored, the loss of orthogonality problem can be corrected by re-orthogonalization [4, 5, 7]. No spurious eigenvalue is generated in this case. However, because each Lanczos step generates one vector, a large amount of computer memory may be required to store all the Lanczos vectors. To limit the maximum amount of memory used, we typically restart the Lanczos algorithm after a certain number of steps. The restarted versions usually use considerably more matrix-vector multiplications than the non-restarted version. In recent years, newly developed restarting strategies have significantly reduced the number of matrix-vector multiplications used. The two most successful ones are the implicit restarting technique [1, 3, 8] and the dynamic thick-restart technique [9, 12]. For symmetric or Hermitian eigenvalue problems, these two schemes are equivalent. Because the thick-restart scheme is easier to implement and it is slightly more flexible than the implicit restarted scheme [9, 12], the new method described here uses the thick-restart scheme. Other thick-restart eigenvalue methods, e.g., the thick-restart Davidson method, can be applied on symmetric eigenvalue problems as well. Compared to them, the main advantage of the new scheme is that it uses less arithmetic operations by taking full advantage of the symmetry of the matrix [13].
Starting Page	1
Ending Page	5
Page Count	5
File Format	HTM / HTML
ISBN	089791984X
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	1998-11-07
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

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