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BIT Numerical Mathematics

, Volume 55, Issue 3, pp 869–896 | Cite as

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

  • Lei-Hong Zhang
  • Wen-Wei Lin
  • Ren-Cang LiEmail author
Article

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.

Keywords

Linear response eigenvalue problem Eigenvalue approximation Rayleigh–Ritz approximation Backward perturbation Error bound Deflating subspace 

Mathematics Subject Classification

65L15 65F15 81Q15 15A18 15A42 

Notes

Acknowledgments

The authors are grateful to the editor and two anonymous referees for their careful reading and helpful comments and suggestions, which have improved the paper considerably. The explicit formulas of \(S_K\) and \(S_M\) and their derivations in Theorem 3.2 are in fact due to one of the referees. Zhang was supported in part by the National Natural Science Foundation of China NSFC-11101257, NSFC-11371102, and the Basic Academic Discipline Program, the 11th five year plan of 211 Project for Shanghai University of Finance and Economics. Part of this work is done while Zhang was a visiting scholar at the Department of Mathematics, University of Texas at Arlington from February 2013 to January 2014. Lin was supported in part by National Center of Theoretical Science, Taiwan and ST Yau Center, Chiao Tung University, Taiwan. Li was supported in part by NSF grants DMS-1115834 and DMS-1317330, by a Research Gift Grant from Intel Corporation, and by National Center of Theoretical Science, Taiwan while he visited in December 2013.

References

  1. 1.
    Bai, Z., Li, R.C.: Minimization principle for linear response eigenvalue problem, I: theory. SIAM J. Matrix Anal. Appl. 33(4), 1075–1100 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Benner, P., Mehrmann, V., Xu, H.: Perturbation analysis for the eigenvalue problem of a formal product of matrices. BIT 42(1), 1–43 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bhatia, R.: Some inequalities for norm ideals. Commun. Math. Phys. 111, 33–39 (1987)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bhatia, R.: Matrix analysis. Graduate texts in mathematics, vol. 169. Springer, New York (1996)Google Scholar
  6. 6.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bhatia, R., Kittaneh, F., Li, R.C.: Eigenvalues of symmetrizable matrices. BIT 38(1), 1–11 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bhatia, R., Li, R.C.: On perturbations of matrix pencils with real spectra. II. Math. Comp. 65(214), 637–645 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Granat, R., Kågström, B., Kressner, D.: Computing periodic deflating subspaces associated with a specified set of eigenvalues. BIT 43(1), 1–18 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Harville, D.A.: Matrix Algebra From a Statistician’s Perspective. Springer, New York (1997)zbMATHCrossRefGoogle Scholar
  13. 13.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)zbMATHCrossRefGoogle Scholar
  14. 14.
    Kahan, W.: Inclusion theorems for clusters of eigenvalues of Hermitian matrices. Computer Science Department, University of Toronoto, Technical report (1967)Google Scholar
  15. 15.
    Kahan, W., Parlett, B.N., Jiang, E.: Residual bounds on approximate eigensystems of nonnormal matrices. SIAM J. Numer. Anal. 19, 470–484 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Kovač-Striko, J., Veselić, K.: Some remarks on the spectra of Hermitian matrices. Linear Algebra Appl. 145, 221–229 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Kressner, D., Pandur, M.M., Shao, M.: An indefinite variant of LOBPCG for definite matrix pencils. Numer. Alg. 66, 681–703 (2014)zbMATHCrossRefGoogle Scholar
  18. 18.
    Krein, M.G.: The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. Mat. Sb. 20, 431–495 (1947)MathSciNetGoogle Scholar
  19. 19.
    Lancaster, P., Ye, Q.: Variational properties and Rayleigh quotient algorithms for symmetric matrix pencils. Oper. Theory: Adv. Appl. 40, 247–278 (1989)MathSciNetGoogle Scholar
  20. 20.
    Li, C.K., Li, R.C.: A note on eigenvalues of perturbed Hermitian matrices. Linear Algebra Appl. 395, 183–190 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Li, R.C.: A perturbation bound for definite pencils. Linear Algebra Appl. 179, 191–202 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Li, R.C.: On perturbations of matrix pencils with real spectra. Math. Comp. 62, 231–265 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Li, R.C.: On perturbations of matrix pencils with real spectra, a revisit. Math. Comp. 72, 715–728 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 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. 25.
    Liang, X., Li, R.C., Bai, Z.: Trace minimization principles for positive semi-definite pencils. Linear Algebra Appl. 438, 3085–3106 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Lin, W.W., Sun, J.G.: Perturbation analysis for the eigenproblem of periodic matrix pairs. Linear Algebra Appl. 337(13), 157–187 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 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)Google Scholar
  28. 28.
    Lu, T.X.: Perturbation bounds of eigenvalues of symmetrizable matrices. Numer. Math. J. Chin. Univ. 16, 177–185 (1994). In ChinesezbMATHGoogle Scholar
  29. 29.
    Mathias, R.: Quadratic residual bounds for the Hermitian eigenvalue problem. SIAM J. Matrix Anal. Appl. 19, 541–550 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Parlett, B.N.: The Symmetric Eigenvalue Problem. SIAM, Philadelphia (1998)zbMATHCrossRefGoogle Scholar
  31. 31.
    Stewart, G.W.: On the sensitivity of the eigenvalue problem \(Ax=\lambda Bx\). SIAM J. Numer. Anal. 4, 669–686 (1972)CrossRefGoogle Scholar
  32. 32.
    Stewart, G.W.: Perturbation bounds for the definite generalized eigenvalue problem. Linear Algebra Appl. 23, 69–86 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Stewart, G.W., Sun, J.G.: Matrix Perturbation Theory. Academic Press, Boston (1990)zbMATHGoogle Scholar
  34. 34.
    Sun, J.G.: A note on Stewart’s theorem for definite matrix pairs. Linear Algebra Appl. 48, 331–339 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Sun, J.G.: Perturbation bounds for eigenspaces of a definite matrix pair. Numer. Math. 41, 321–343 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Sun, J.G.: Backward perturbation analysis of certain characteristic subspaces. Numer. Math. 65, 357–382 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Sun, J.G.: A note on backward perturbations for the Hermitian eigenvalue problem. BIT 35, 385–393 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  38. 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)Google Scholar
  39. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 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. 41.
    Thouless, D.J.: Vibrational states of nuclei in the random phase approximation. Nucl. Phys. 22(1), 78–95 (1961)zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Thouless, D.J.: The quantum mechanics of many-body systems. Academic Press, New York (1972)Google Scholar
  43. 43.
    Tsiper, E.V.: Variational procedure and generalized Lanczos recursion for small-amplitude classical oscillations. JETP Lett. 70(11), 751–755 (1999)CrossRefGoogle Scholar
  44. 44.
    Tsiper, E.V.: A classical mechanics technique for quantum linear response. J. Phys. B At. Mol. Opt. Phys. 34(12), L401–L407 (2001)CrossRefGoogle Scholar
  45. 45.
    van Hemmen, J.L., Ando, T.: An inequality for trace ideals. Commun. Math. Phys. 76, 143–148 (1980)zbMATHCrossRefGoogle Scholar
  46. 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)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.School of MathematicsShanghai University of Finance and EconomicsShanghaiPeople’s Republic of China
  2. 2.Department of Applied MathematicsNational Chiao Tung UniversityHsinchuTaiwan
  3. 3.Department of MathematicsUniversity of Texas at ArlingtonArlingtonUSA

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