Filters consist of a few resolvents to solve real symmetric definite generalized eigenproblems

  • Hiroshi MurakamiEmail author
Special Feature: Original Paper International Workshop on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing (EPASA2018)


By using a filter, we solve those eigenpairs of a real symmetric definite generalized eigenproblem \(A{\mathbf {v}}=\lambda B{\mathbf {v}}\) whose eigenvalues are in a specified real interval. In present study, the filter is a polynomial of the real-part of a linear combination of a few resolvents, and the polynomial is restricted to a Chebyshev polynomial to make the design of the filter simple. In order to apply a few resolvents, the same number of systems of linear equations with different shifts are solved. In present study, we assume those systems of linear equations are solved by some direct method using matrix factorizations. Since only a few resolvents are used, the number of required factorizations is also a few (2–4).


Filter Eigenproblem Resolvent Polynomial 

Mathematics Subject Classification

94A12 65F15 47A10 



  1. 1.
    Austin, A.P., Trefethen, L.N.: Computing eigenvalues of real symmetric matrices with rational filters in real arithmetic. SIAM J. Sci. Comput 37(3), A1365–A1387 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Daniels, R.W.: Approximation Methods for Electronic Filter Design. McGraw-Hill, New York (1974)Google Scholar
  3. 3.
    Demmel, J., Veselić, K.: Jacobi’s method is more accurate than \(QR\). SIAM J. Matrix Anal. Appl. 13(4), 1204–1245 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Galgon, M., Krämer, L., Lang, B.: The FEAST algorithm for large eigenvalue problems. PAMM 11, 747–748 (2011)CrossRefGoogle Scholar
  5. 5.
    Güttel, S., Polizzi, P., Tang, P.T.P., Viaud, G.: Zolotarev quadrature rules and load balancing for the FEAST eigensolver. SIAM J. Sci. Comput 37(4), A2100–A2122 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ikegami, T., Sakurai, T., Nagashima, U.: A filter diagonalization for generalized eigenvalue problems based on the Sakurai–Sugiura projection method. J. Compu. Appl. Math. 233(8), 1927–1936 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lutovac, M.D., Tos̆ić, D.V., Evans, B.L.: Filter Design for Signal Processing. §12.8. Prentice Hall, Englewood Cliffs (2001)Google Scholar
  8. 8.
    Murakami, H.: Filter diagonalization method by using a polynomial of a resolvent as the filter for a real symmetric-definite generalized eigenproblem. In: Sakurai, T., Zhang, S.L., Imamura, T., Yamamoto, Y., Kuramashi, Y., Hoshi, T.(eds.) Proceedings of EPASA2015, Tsukuba, Japan, Sept. 2015, pp. 205–232. Springer LNCSE-117 (2018)Google Scholar
  9. 9.
    Polizzi, E.: A density matrix-based algorithm for solving eigenvalue problems. Phys. Rev. B 79(1), 115112–115117 (2009)CrossRefGoogle Scholar
  10. 10.
    Rutishauser, H.: The Jacobi method for real symmetric matrices. Numer. Math. 9(1), 1–10 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sakurai, T., Sugiura, H.: A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math. 159, 119–128 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sakurai, T., Tadano, H.: CIRR: a Rayleigh–Ritz type method with contour integral for generalized eigenvalue problems. Hokkaido Math. J. 36(4), 745–757 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Toledo, S., Rabani, E.: Very large electronic structure calculations using an out-of-core filter-diagonalization method. J. Comput. Phys. 180(1), 256–269 (2002)CrossRefzbMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Tokyo Metropolitan UniversityTokyoJapan

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