nach oben

BIT Numerical Mathematics

Erschienen in:

01.09.2015

Backward perturbation analysis and residual-based error bounds for the linear response eigenvalue problem

verfasst von: Lei-Hong Zhang, Wen-Wei Lin, Ren-Cang Li

Erschienen in: BIT Numerical Mathematics | Ausgabe 3/2015

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

The numerical solution of a large scale linear response eigenvalue problem is often accomplished by computing a pair of deflating subspaces associated with the interesting part of the spectrum. This paper is concerned with the backward perturbation analysis for a given pair of approximate deflating subspaces or an approximate eigenquadruple. Various optimal backward perturbation bounds are obtained, as well as bounds for approximate eigenvalues computed through the pair of approximate deflating subspaces or approximate eigenquadruple. These results are reminiscent of many existing classical ones for the standard eigenvalue problem.

Vorheriger Artikel A multi-level spectral deferred correction method

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

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!

Jetzt informieren

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!

Jetzt informieren

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!

Jetzt informieren

Conceivably, there are other possible ones that are equally appropriate. For example, one may define another optimal backward error by replacing \(\left\| \Delta K\right\| _{{{\mathrm{ui}}}}+\left\| \Delta M\right\| _{{{\mathrm{ui}}}}\) in (3.8) by \(\sqrt{\left\| \Delta K\right\| _{{{\mathrm{ui}}}}^2+\left\| \Delta M\right\| _{{{\mathrm{ui}}}}^2}\). Such \(\zeta \) differs from (3.8) within a constant factor.

For a function \(f\) with a matrix argument \(X\in \mathbb {R}^{n\times m}\), its partial derivative \(\frac{\partial f(X)}{\partial X}\in \mathbb {R}^{n\times m}\) with its \({\scriptstyle (i,j)}\)th entry \(\frac{\partial f(X)}{\partial X_{(i,j)}}\) [12, Chapter 15].

This idea of turning (3.15a) into the more “friendly” (3.16) is due to one of the referees.

This is actually true even for \(K\succeq 0\) and \(M\succeq 0\). But in stating Theorem 4.3, we stick to our default assumption on \(K\) and \(M\) just for consistency.

Bai, Z., Li, R.C.: Minimization principle for linear response eigenvalue problem, I: theory. SIAM J. Matrix Anal. Appl. 33(4), 1075–1100 (2012)MATHMathSciNetCrossRef

Bai, Z., Li, R.C.: Minimization principles for the linear response eigenvalue problem II: computation. SIAM J. Matrix Anal. Appl. 34(2), 392–416 (2013)MATHMathSciNetCrossRef

Benner, P., Mehrmann, V., Xu, H.: Perturbation analysis for the eigenvalue problem of a formal product of matrices. BIT 42(1), 1–43 (2002)MATHMathSciNetCrossRef

Bhatia, R.: Some inequalities for norm ideals. Commun. Math. Phys. 111, 33–39 (1987)MATHCrossRef

Bhatia, R.: Matrix analysis. Graduate texts in mathematics, vol. 169. Springer, New York (1996)

Bhatia, R., Kittaneh, F., Li, R.C.: Some inequalities for commutators and an application to spectral variation. II. Lin. Multilin. Alg. 43(1–3), 207–220 (1997)MATHMathSciNetCrossRef

Bhatia, R., Kittaneh, F., Li, R.C.: Eigenvalues of symmetrizable matrices. BIT 38(1), 1–11 (1998)MATHMathSciNetCrossRef

Bhatia, R., Li, R.C.: On perturbations of matrix pencils with real spectra. II. Math. Comp. 65(214), 637–645 (1996)MATHMathSciNetCrossRef

Cao, Z.H., Xie, J.J., Li, R.C.: A sharp version of Kahan’s theorem on clustered eigenvalues. Linear Algebra Appl. 245, 147–155 (1996)MATHMathSciNetCrossRef

10.

Dzeng, D.C., Lin, W.W.: Homotopy continuation method for the numerical solutions of generalised symmetric eigenvalue problems. J. Austral. Math. Soc. Ser. B 32, 437–456 (1991)MATHMathSciNetCrossRef

11.

Granat, R., Kågström, B., Kressner, D.: Computing periodic deflating subspaces associated with a specified set of eigenvalues. BIT 43(1), 1–18 (2003)MathSciNetCrossRef

12.

Harville, D.A.: Matrix Algebra From a Statistician’s Perspective. Springer, New York (1997)MATHCrossRef

13.

Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)MATHCrossRef

14.

Kahan, W.: Inclusion theorems for clusters of eigenvalues of Hermitian matrices. Computer Science Department, University of Toronoto, Technical report (1967)

15.

Kahan, W., Parlett, B.N., Jiang, E.: Residual bounds on approximate eigensystems of nonnormal matrices. SIAM J. Numer. Anal. 19, 470–484 (1982)MATHMathSciNetCrossRef

16.

Kovač-Striko, J., Veselić, K.: Some remarks on the spectra of Hermitian matrices. Linear Algebra Appl. 145, 221–229 (1991)MATHMathSciNetCrossRef

17.

Kressner, D., Pandur, M.M., Shao, M.: An indefinite variant of LOBPCG for definite matrix pencils. Numer. Alg. 66, 681–703 (2014)MATHCrossRef

18.

Krein, M.G.: The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. Mat. Sb. 20, 431–495 (1947)MathSciNet

19.

Lancaster, P., Ye, Q.: Variational properties and Rayleigh quotient algorithms for symmetric matrix pencils. Oper. Theory: Adv. Appl. 40, 247–278 (1989)MathSciNet

20.

Li, C.K., Li, R.C.: A note on eigenvalues of perturbed Hermitian matrices. Linear Algebra Appl. 395, 183–190 (2005)MATHMathSciNetCrossRef

21.

Li, R.C.: A perturbation bound for definite pencils. Linear Algebra Appl. 179, 191–202 (1993)MATHMathSciNetCrossRef

22.

Li, R.C.: On perturbations of matrix pencils with real spectra. Math. Comp. 62, 231–265 (1994)MATHMathSciNetCrossRef

23.

Li, R.C.: On perturbations of matrix pencils with real spectra, a revisit. Math. Comp. 72, 715–728 (2003)MATHMathSciNetCrossRef

24.

Liang, X., Li, R.C.: The hyperbolic quadratic eigenvalue problem. Technical Report 2014–01, Department of Mathematics, University of Texas at Arlington. www.uta.edu/math/preprint/ (2014)

25.

Liang, X., Li, R.C., Bai, Z.: Trace minimization principles for positive semi-definite pencils. Linear Algebra Appl. 438, 3085–3106 (2013)MATHMathSciNetCrossRef

26.

Lin, W.W., Sun, J.G.: Perturbation analysis for the eigenproblem of periodic matrix pairs. Linear Algebra Appl. 337(13), 157–187 (2001)MATHMathSciNetCrossRef

27.

Lin, W.W., van Dooren, P., Xu, Q.F.: Equivalent characterizations of periodical invariant subspaces. NCTS Preprints Series 1998–8, National Center for Theoretical Sciences, Math. Division, National Tsing Hua University, Hsinchu, Taiwan (1998)

28.

Lu, T.X.: Perturbation bounds of eigenvalues of symmetrizable matrices. Numer. Math. J. Chin. Univ. 16, 177–185 (1994). In ChineseMATH

29.

Mathias, R.: Quadratic residual bounds for the Hermitian eigenvalue problem. SIAM J. Matrix Anal. Appl. 19, 541–550 (1998)MATHMathSciNetCrossRef

30.

Parlett, B.N.: The Symmetric Eigenvalue Problem. SIAM, Philadelphia (1998)MATHCrossRef

31.

Stewart, G.W.: On the sensitivity of the eigenvalue problem \(Ax=\lambda Bx\). SIAM J. Numer. Anal. 4, 669–686 (1972)CrossRef

32.

Stewart, G.W.: Perturbation bounds for the definite generalized eigenvalue problem. Linear Algebra Appl. 23, 69–86 (1979)MATHMathSciNetCrossRef

33.

Stewart, G.W., Sun, J.G.: Matrix Perturbation Theory. Academic Press, Boston (1990)MATH

34.

Sun, J.G.: A note on Stewart’s theorem for definite matrix pairs. Linear Algebra Appl. 48, 331–339 (1982)MATHMathSciNetCrossRef

35.

Sun, J.G.: Perturbation bounds for eigenspaces of a definite matrix pair. Numer. Math. 41, 321–343 (1983)MATHMathSciNetCrossRef

36.

Sun, J.G.: Backward perturbation analysis of certain characteristic subspaces. Numer. Math. 65, 357–382 (1993)MATHMathSciNetCrossRef

37.

Sun, J.G.: A note on backward perturbations for the Hermitian eigenvalue problem. BIT 35, 385–393 (1995)MATHMathSciNetCrossRef

38.

Sun, J.G.: Stability and accuracy: perturbation analysis of algebraic eigenproblems. Technical report UMINF 1998–07, ISSN-0348-0542, Faculty of Science and Technology, Department of Computing Science, Umeå University (1998)

39.

Teng, Z., Li, R.C.: Convergence analysis of Lanczos-type methods for the linear response eigenvalue problem. J. Comput. Appl. Math. 247, 17–33 (2013)MATHMathSciNetCrossRef

40.

Teng, Z., Zhou, Y., Li, R.C.: A block Chebyshev-Davidson method for linear response eigenvalue problems. Technical Report 2013–11, Department of Mathematics, University of Texas at Arlington. www.uta.edu/math/preprint/ (2013)

41.

Thouless, D.J.: Vibrational states of nuclei in the random phase approximation. Nucl. Phys. 22(1), 78–95 (1961)MATHMathSciNetCrossRef

42.

Thouless, D.J.: The quantum mechanics of many-body systems. Academic Press, New York (1972)

43.

Tsiper, E.V.: Variational procedure and generalized Lanczos recursion for small-amplitude classical oscillations. JETP Lett. 70(11), 751–755 (1999)CrossRef

44.

Tsiper, E.V.: A classical mechanics technique for quantum linear response. J. Phys. B At. Mol. Opt. Phys. 34(12), L401–L407 (2001)CrossRef

45.

van Hemmen, J.L., Ando, T.: An inequality for trace ideals. Commun. Math. Phys. 76, 143–148 (1980)MATHCrossRef

46.

Zhang, L.H., Xue, J., Li, R.C.: Rayleigh-Ritz approximation for the linear response eigenvalue problem. SIAM J. Matrix Anal. Appl. 35, 765–782 (2014)MATHMathSciNetCrossRef

Titel: Backward perturbation analysis and residual-based error bounds for the linear response eigenvalue problem
verfasst von: Lei-Hong Zhang
Wen-Wei Lin
Ren-Cang Li
Publikationsdatum: 01.09.2015
Verlag: Springer Netherlands
Erschienen in: BIT Numerical Mathematics / Ausgabe 3/2015
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-014-0519-8

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Springer Professional "Wirtschaft+Technik"

Weitere Artikel der Ausgabe 3/2015

Monotone iterative ADI method for semilinear parabolic problems

Superconvergent cubic spline quasi-interpolants on Powell-Sabin partitions

Efficient algorithm for simultaneous reduction to the -Hessenberg-triangular-triangular form

Spline element method for Monge–Ampère equations

Axel Ruhe 1942–2015

Between moving least-squares and moving least-

Premium Partner