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A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

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Abstract

The paper discusses the following topics: attractions of the real tridiagonal case, relative eigenvalue condition number for matrices in factored form, dqds, triple dqds, error analysis, new criteria for splitting and deflation, eigenvectors of the balanced form, twisted factorizations and generalized Rayleigh quotient iteration. We present our fast real arithmetic algorithm and compare it with alternative published approaches.

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References

  1. Bini, D.A., Gemignani, L., Tisseur, F.: The Ehrlich–Aberth method for the nonsymmetric tridiagonal eigenvalue problem. SIAM J. Matrix Anal. Appl. 27(1), 153–175 (2005)

    Article  MathSciNet  Google Scholar 

  2. Bunse-Gerstner, A.: An analysis of the HR algorithm for computing the eigenvalues of a matrix. Linear Algebra Appl. 35, 155–173 (1981)

    Article  MathSciNet  Google Scholar 

  3. Clement, P.A.: A class of triple-diagonal matrices for test purposes. In: SIAM Review, vol. 1 (1959)

  4. Cullum, J.K.: A QL procedure for computing the eigenvalues of complex symmetric tridiagonal matrices. SIAM J. Matrix Anal. Appl. 17(1), 83–109 (1996)

    Article  MathSciNet  Google Scholar 

  5. Day, D.: Semi-duality in the Two-sided Lanczos Algorithm. Ph.D Thesis, University of California, Berkeley (1993)

  6. Demmel, J.W.: Applied Numerical Linear Algebra, Society for Industrial and Applied Mathematics (1997)

  7. Dhillon, I.S., Parlett, B.N.: Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices. Linear Algebra Appl. 387, 1–28 (2004)

    Article  MathSciNet  Google Scholar 

  8. Dhillon, I.S., Parlett, B.N.: Orthogonal eigenvectors and relative gaps. SIAM J. Matrix Anal. Appl. 25, 858–899 (2004)

    Article  MathSciNet  Google Scholar 

  9. Fernando, K.V., Parlett, B.: Accurate singular values and differential QD algorithms. Numer. Math. 67, 191–229 (1994)

    Article  MathSciNet  Google Scholar 

  10. Fernando, K.V.: On computing an eigenvector of a tridiagonal matrix. Part I: basic results. SIAM J. Matrix Anal. Appl. 18, 1013–1034 (1997)

    Article  MathSciNet  Google Scholar 

  11. Ferreira, C.: The Unsymmetric Tridiagonal Eigenvalue Problem. Ph.D Thesis, University of Minho (2007). http://hdl.handle.net/1822/6761

  12. Ferreira, C., Parlett, B.: Convergence of LR algorithm for a one-point spectrum tridiagonal matrix. Numer. Math. 113(3), 417–431 (2009)

    Article  MathSciNet  Google Scholar 

  13. Ferreira, C., Parlett, B., Froilán, M.D.: Sensitivity of eigenvalues of an unsymmetric tridiagonal matrix. Numer. Math. (2012). https://doi.org/10.1007/s00211-012-0470-z

    Article  MathSciNet  MATH  Google Scholar 

  14. Francis, J.G.F.: The QR transformation—a unitary analogue to the LR transformation, parts I and II. Comput. J. 4, 265–272; 332–245 (1961/1962)

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

    Article  MathSciNet  Google Scholar 

  16. Liu, Z.A.: On the extended HR algorithm. Technical Report PAM-564, Center for Pure and Applied Mathematics, University of California, Berkeley, CA, USA (1992)

  17. Parlett, B.N., Reinsch, C.: Balancing a matrix for calculation of eigenvalues and eigenvectors. Numer. Math. 13, 292–304 (1969)

    Article  Google Scholar 

  18. Parlett, B.N.: The Rayleigh quocient iteration and some generalizations for non-normal matrices. Math. Comput. 28(127), 679–693 (1974)

    Article  Google Scholar 

  19. Parlett, B.N.: The contribution of J. H. Wilkinson to numerical analysis. In: Nash, S.G. (ed.), A History of Scientific Computing, ACM Press, p. 25 (1990)

  20. Parlett, B.N.: Reduction to tridiagonal form and minimal realizations. SIAM J. Matrix Anal. Appl. 13, 567–593 (1992)

    Article  MathSciNet  Google Scholar 

  21. Parlett, B.N.: The new QD algorithms. Acta Numer 4, 459–491 (1995)

    Article  MathSciNet  Google Scholar 

  22. Parlett, B.N., Dhillon, I.S.: Fernandos solution to Wilkinsons problem: an application of double factorization. Linear Algebra Appl. 267, 247–279 (1997)

    Article  MathSciNet  Google Scholar 

  23. Parlett, B.N., Marques, O.A.: An implementation of the DQDS algorithm. Linear Algebra Appl. 309, 217–259 (2000)

    Article  MathSciNet  Google Scholar 

  24. Parlett, B., Dopico, F.M., Ferreira, C.: The inverse eigenvector problem for real tridiagonal matrices. SIAM J. Matrix Anal. Appl. 37, 577–597 (2016)

    Article  MathSciNet  Google Scholar 

  25. Pasquini, L.: Accurate computation of the zeros of the generalized Bessel polynomials. Numerische Mathematic 86, 507–538 (2000)

    Article  MathSciNet  Google Scholar 

  26. Rutishauser, H.: Der Quotienten-Differenzen-Algorithmus. Z. Angew. Math. Physik 5, 233–251 (1954)

    Article  MathSciNet  Google Scholar 

  27. Rutishauser, H.: Der Quotienten-Differenzen-Algorithmus. Mitt. Inst. Angew. Math. ETH, vol. 7, Birkh\(\ddot{\text{a}}\)user, Basel (1957)

  28. Rutishauser, H.: Solution of eigenvalue problems with the LR-transformation. Natl. Bur. Stand. Appl. Math. Ser. 49, 47–81 (1958)

    MathSciNet  MATH  Google Scholar 

  29. Rutishauser, H., Schwarz, H.R.: The LR transformation method for symmetric matrices. Numer. Math. 5, 273–289 (1963)

    Article  MathSciNet  Google Scholar 

  30. Slemons, J.: Toward the Solution of the Eigenproblem: Nonsymmetric Tridiagonal Matrices. Ph.D Thesis, University of Washington, Seattle (2008)

  31. Slemons, J.: The result of two steps of the LR algorithm is diagonally similar to the result of one step of the HR algorithm. SIAM J. Matrix Anal. Appl. 31(1), 68–74 (2009)

    Article  MathSciNet  Google Scholar 

  32. Trefethen, L.N., Embree, M.: Spectra and pseudospectra. In: The Behavior of Nonnormal Matrices and Operators, Princeton University Press (2005)

  33. Watkins, D.S.: QR-like algorithms—an overview of convergence theory and practice. Lect. Appl. Math. 32, 879–893 (1996)

    MathSciNet  MATH  Google Scholar 

  34. Watkins, D.S., Elsner, L.: Convergence of algorithms of decomposition type for the eigenvalue problem. Linear Algebra Appl. 143, 19–47 (1991)

    Article  MathSciNet  Google Scholar 

  35. Willems, P.R., Lang, B.: Twisted factorizations and QD-type transformations for the \(\text{ MR}^3\) algorithm—new representations and analysis. SIAM J. Matrix Anal. Appl. 33(2), 523–553 (2012)

    Article  MathSciNet  Google Scholar 

  36. Wu. Z.: The Triple DQDS Algorithm for Complex Eigenvalues. Ph.D Thesis, University of California, Berkeley (1996)

  37. Xu, H.: The relation between the QR and LR algorithms. SIAM J. Matrix Anal. Appl. 19(2), 551–555 (1998)

    Article  MathSciNet  Google Scholar 

  38. Yao, Y.: Error Analysis of the QDs and DQDs Algorithms. Ph.D Thesis, University of California, Berkeley (1994)

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Acknowledgements

The authors would like to thank Associate Editor Martin H. Gutknecht and the anonymous referees for forcing us to look more deeply into an error analysis of our triple dqds algorithm (first version) and to give a clearer presentation of its mathematical analysis and implemention details (last version).

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Authors and Affiliations

  1. Centro de Matemática, Universidade do Minho, 4710-057, Braga, Portugal

    Carla Ferreira

  2. Division of the Electrical Engineering and Computer Science, Department of Mathematics and the Computer Science, University of California, Berkeley, CA, 94720, USA

    Beresford Parlett

Authors
  1. Carla Ferreira
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  2. Beresford Parlett
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Corresponding author

Correspondence to Carla Ferreira.

Additional information

The research of the first author was partially financed by Portuguese Funds through FCT (Fundação para a Ciência e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020.

Appendices

3dqds algorithm

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Pseudocode for the whole algorithm

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Ferreira, C., Parlett, B. A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem. Numer. Math. 150, 373–422 (2022). https://doi.org/10.1007/s00211-021-01254-z

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  • DOI: https://doi.org/10.1007/s00211-021-01254-z

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