Top

Journal of Scientific Computing

Published in:

26-09-2018

A Contour-Integral Based Method for Counting the Eigenvalues Inside a Region

Author: Guojian Yin

Published in: Journal of Scientific Computing | Issue 3/2019

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

previous article Lowest-Order Weak Galerkin Finite Element Methods for Linear Elasticity on Rectangular and Brick Meshes

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Ahlfors, L.: Complex Analysis, 3rd edn. McGraw-Hill, Inc., New York City (1979)MATH

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)CrossRefMATH

Bai, Z., Demmel, J., Gu, G.: An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblem. Numer. Math. 76, 279–308 (1997)MathSciNetCrossRefMATH

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)CrossRef

Cerdá, J., Soria, F.: Accurate and transferable extended Hückel-type tight-binding parameters. Phys. Rev. B 61, 7965 (2000)CrossRef

Chan, T.T.: Rank revealing QR factorizations. Linear Algebra Appl. 88–89, 67–82 (1987)MathSciNetMATH

Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, Orlando (1984)MATH

Demmel, J.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)CrossRefMATH

Di Napoli, E., Polizzi, E., Saad, Y.: Efficient estimation of eigenvalue counts in an interval. arXiv:1308.4275

10.

Futamura, Y., Tadano, H., Sakurai, T.: Parallel stochastic estimation method of eigenvalue distribution. JSIAM Lett. 2, 127–130 (2010)MathSciNetCrossRefMATH

11.

Gantmacher, F.R.: The Theory of Matrices. Chelsea, New York (1959)MATH

12.

Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)MATH

13.

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)MathSciNetCrossRefMATH

14.

Hinrichsen, D., Pritchard, A.J.: Mathematical Systems Theory I: Modelling, State Space Analysis and Robustness. Springer, Berlin (2005)CrossRefMATH

15.

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)CrossRef

16.

Kamgnia, E.R., Philippe, B.: Counting eigenvalues in domains of the complex field. arXiv:1110.4797

17.

Lin, L., Saad, Y., Yang, C.: Approximating spectral densities of large matrices. arXiv:1308.5467

18.

Moler, C.B., Stewart, G.W.: An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10, 241–256 (1973)MathSciNetCrossRefMATH

19.

Polizzi, E.: Density-matrix-based algorithm for solving eigenvalue problems. Phys. Rev. B 79, 115112 (2009)CrossRef

20.

Ruhe, A.: Rational Krylov: a practical algorithm for large sparse nonsymmetric matrix pencils. SIAM J. Sci. Comput. 19, 1535–1551 (1998)MathSciNetCrossRefMATH

21.

Saad, Y.: Numerical Methods for Large Eigenvalue Problems. SIAM, Philadelphia (2011)CrossRefMATH

22.

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)MathSciNetCrossRef

23.

Sakurai, T., Sugiura, H.: A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math. 159, 119–128 (2003)MathSciNetCrossRefMATH

24.

Sakurai, T., Tadano, H.: CIRR: a Rayleigh–Ritz type method with contour integral for generalized eigenvalue problems. Hokkaido Math. J. 36, 745–757 (2007)MathSciNetCrossRefMATH

25.

Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edn. Springer, New York (1998)CrossRefMATH

26.

Stewart, G.W.: Matrix Algorithms, Vol. II, Eigensystems. SIAM, Philadelphia (2001)CrossRefMATH

27.

Tang, P., Polizzi, E.: FEAST as a subspace iteration eigensolver accelerated by approximate spectral projection. SIAM J. Matrix Anal. Appl. 35, 354–390 (2014)MathSciNetCrossRefMATH

28.

von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17, 395–416 (2007)MathSciNetCrossRef

29.

Yin, G., Chan, R., Yueng, M.-C.: A FEAST algorithm with oblique projection for generalized eigenvalue problems. Numer. Linear Algebra Appl. 24, e2092 (2017)MathSciNetCrossRefMATH

Title: A Contour-Integral Based Method for Counting the Eigenvalues Inside a Region
Author: Guojian Yin
Publication date: 26-09-2018
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2019
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0838-z

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 3/2019

Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation

The Jump Start Power Method: A New Approach for Computing the Ergodic Projector of a Finite Markov Chain

Correction to: A Fourth-Order Kernel-Free Boundary Integral Method for the Modified Helmholtz Equation

Variational Models for Joint Subsampling and Reconstruction of Turbulence-Degraded Images

Lowest-Order Weak Galerkin Finite Element Methods for Linear Elasticity on Rectangular and Brick Meshes

Analysis of Fully Discrete Approximations for Dissipative Systems and Application to Time-Dependent Nonlocal Diffusion Problems

Premium Partner