Abstract
In many applications, the information about the number of eigenvalues inside a given region is required. In this work, we develop a contour-integral based method for this purpose. Our method is motivated by two findings. There exist methods for estimating the number of eigenvalues inside a region in the complex plane, but our method is able to compute the number exactly. Our method has a good potential to be implemented on a high-performance parallel architecture. Numerical experiments are reported to show the viability of our method.
Similar content being viewed by others
References
Ahlfors, L.: Complex Analysis, 3rd edn. McGraw-Hill, Inc., New York City (1979)
Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., Van der Vorst, H.: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia (2000)
Bai, Z., Demmel, J., Gu, G.: An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblem. Numer. Math. 76, 279–308 (1997)
Boisvert, R.F., Pozo, R., Remington, K., Barrett, R., Dongarra, J.: The matrix market: a web resource for test matrix collections. In: Boisvert, R.F. (ed.) Quality of Numerical Software: Assessment and Enhancement, IFIP Advances in Information and Communication Technology, pp. 125–137. Chapman & Hall, London (1997)
Cerdá, J., Soria, F.: Accurate and transferable extended Hückel-type tight-binding parameters. Phys. Rev. B 61, 7965 (2000)
Chan, T.T.: Rank revealing QR factorizations. Linear Algebra Appl. 88–89, 67–82 (1987)
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, Orlando (1984)
Demmel, J.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)
Di Napoli, E., Polizzi, E., Saad, Y.: Efficient estimation of eigenvalue counts in an interval. arXiv:1308.4275
Futamura, Y., Tadano, H., Sakurai, T.: Parallel stochastic estimation method of eigenvalue distribution. JSIAM Lett. 2, 127–130 (2010)
Gantmacher, F.R.: The Theory of Matrices. Chelsea, New York (1959)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
Grimes, R.G., Lewis, J.D., Simon, H.D.: A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems. SIAM J. Matrix Anal. Appl. 15, 228–272 (1994)
Hinrichsen, D., Pritchard, A.J.: Mathematical Systems Theory I: Modelling, State Space Analysis and Robustness. Springer, Berlin (2005)
Hoshi, T., Yamamoto, S., Fujiwara, T., Sogabe, T., Zhang, S.L.: An order-\(N\) electronic structure theory with generalized eigenvalue equations and its application to a ten-million-atom system. J. Phys. Condens. Matter 24, 165502 (2012)
Kamgnia, E.R., Philippe, B.: Counting eigenvalues in domains of the complex field. arXiv:1110.4797
Lin, L., Saad, Y., Yang, C.: Approximating spectral densities of large matrices. arXiv:1308.5467
Moler, C.B., Stewart, G.W.: An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10, 241–256 (1973)
Polizzi, E.: Density-matrix-based algorithm for solving eigenvalue problems. Phys. Rev. B 79, 115112 (2009)
Ruhe, A.: Rational Krylov: a practical algorithm for large sparse nonsymmetric matrix pencils. SIAM J. Sci. Comput. 19, 1535–1551 (1998)
Saad, Y.: Numerical Methods for Large Eigenvalue Problems. SIAM, Philadelphia (2011)
Sakurai, T., Futamura, Y., Tadano, H.: Efficient parameter estimation and implementation of a contour integral-based eigensolver. J. Algorithms Comput. Technol. 7, 249–269 (2013)
Sakurai, T., Sugiura, H.: A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math. 159, 119–128 (2003)
Sakurai, T., Tadano, H.: CIRR: a Rayleigh–Ritz type method with contour integral for generalized eigenvalue problems. Hokkaido Math. J. 36, 745–757 (2007)
Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edn. Springer, New York (1998)
Stewart, G.W.: Matrix Algorithms, Vol. II, Eigensystems. SIAM, Philadelphia (2001)
Tang, P., Polizzi, E.: FEAST as a subspace iteration eigensolver accelerated by approximate spectral projection. SIAM J. Matrix Anal. Appl. 35, 354–390 (2014)
von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17, 395–416 (2007)
Yin, G., Chan, R., Yueng, M.-C.: A FEAST algorithm with oblique projection for generalized eigenvalue problems. Numer. Linear Algebra Appl. 24, e2092 (2017)
Acknowledgements
I would like to thank Professor Raymond H. Chan, my thesis advisor, for his help in preparing this paper. I also would like to thank the anonymous reviewers for their useful suggestions which have greatly improved this paper. This work was supported by the National Natural Science Foundation of China (NSFC) under Grant 11701593.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yin, G. A Contour-Integral Based Method for Counting the Eigenvalues Inside a Region. J Sci Comput 78, 1942–1961 (2019). https://doi.org/10.1007/s10915-018-0838-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-018-0838-z