On a Generalized Companion Matrix Pencil for Matrix Polynomials Expressed in the Lagrange Basis

  • Robert M. Corless
Part of the Trends in Mathematics book series (TM)

Abstract

Experimental observations of univariate rootfinding by generalized companion matrix pencils expressed in the Lagrange basis show that the method can sometimes be numerically stable. It has recently been proved that a new rootfinding condition number, defined for points on a set containing the interpolation points, is never larger than the rootfinding condition number for the Bernstein polynomial (which is itself optimally small in a certain sense); and computation shows that sometimes it can be much smaller. These results extend to the matrix polynomial case, although in this case we are not computing polynomial roots but rather ‘polynomial eigenvalues’ (sometimes known as ‘latent roots’), i.e. finding the values of x where the matrix polynomial is singular. This paper gives two theorems explaining part of the influence of the geometry of the interpolation nodes on the conditioning of the rootfinding and eigenvalue problems.

Keywords

Lagrange basis companion matrix polynomial eigenvalue 

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References

  1. [1]
    A. Amiraslani. Dividing polynomials when you only know their values. In Laureano Gonzalez-Vega and Tomas Recio, editors, Proceedings EACA, pages 5–10, June 2004.Google Scholar
  2. [2]
    A. Amiraslani. Algorithms for Matrices, Polynomials, and Matrix Polynomials. PhD thesis, University of Western Ontario, London, Canada, May 2006.Google Scholar
  3. [3]
    A. Amiraslani, D. Aruliah, and R.M. Corless. The Rayleigh quotient iteration for generalized companion matrix pencils. in preparation, 2006.Google Scholar
  4. [4]
    A. Amiraslani, D. Aruliah, R.M. Corless, and N. Rezvani. Pseudospectra of matrix polynomials in different bases. Poster TR-06-03, Ontario Research Centre for Computer Algebra, http://www.orcca.on.ca/techreports, June 2006. presented as a poster at CAIMS/MITACS York, June 2006.Google Scholar
  5. [5]
    A. Amiraslani, R.M. Corless, L. Gonzalez-Vega, and A. Shakoori. Polynomial algebra by values. Technical Report TR-04-01, Ontario Research Centre for Computer Algebra, http://www.orcca.on.ca/TechReports, January 2004.Google Scholar
  6. [6]
    J.-P. Berrut and L.N. Trefethen. Barycentric Lagrange interpolation. SIAM Review, 46(3): 501–517, 2004.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    D.A. Bini and L. Gemignani. Bernstein-Bezoutian matrices. Theor. Comput. Sci., 315(2–3): 319–333, 2004.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    D.A. Bini, L. Gemignani, and V.Y. Pan. Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations. Numer. Math., 100: 373–408, 2005.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    R.M. Corless. Generalized companion matrices in the Lagrange basis. In Laureano Gonzalez-Vega and Tomas Recio, editors, Proceedings EACA, pages 317–322, June 2004.Google Scholar
  10. [10]
    R.M. Corless. The reducing subspace at infinity for the generalized companion matrix in the Lagrange basis. in preparation, 2006.Google Scholar
  11. [11]
    R.M. Corless and S.M. Watt. Bernstein bases are optimal, but, sometimes, Lagrange bases are better. In Proceedings SYNASC, Timisoara, pages 141–153. MITRON Press, September 2004.Google Scholar
  12. [12]
    A. Edelman and H. Murakami. Polynomial roots from companion matrix eigenvalues. Mathematics of Computation, 64(210): 763–776, April 1995.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    R.T. Farouki and T.N.T. Goodman. On the optimal stability of the Bernstein basis. Math. Comput., 65(216): 1553–1566, 1996.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    R.T. Farouki and V.T. Rajan. On the numerical condition of polynomials in Bernstein form. Comput. Aided Geom. Des., 4(3): 191–216, 1987.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    R.T. Farouki and V.T. Rajan. Algorithms for polynomials in Bernstein form. Comput. Aided Geom. Des., 5(1): 1–26, 1988.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    S. Fortune. Polynomial root finding using iterated eigenvalue computation. In Bernard Mourrain, editor, Proceedings ISSAC, pages 121–128, London, Canada, 2001. ACM Press.Google Scholar
  17. [17]
    T. Hermann. On the stability of polynomial transformations between Taylor, Bézier, and Hermite forms. Numerical Algorithms, 13: 307–320, 1996.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    N.J. Higham. Accuracy and Stability of Numerical Algorithms. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, second edition, 2002.MATHGoogle Scholar
  19. [19]
    N.J. Higham. The numerical stability of barycentric Lagrange interpolation. IMA Journal of Numerical Analysis, 24: 547–556, 2004.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    T.E. Hull and R. Mathon. The mathematical basis and a prototype implementation of a new polynomial rootfinder with quadratic convergence. ACM Trans. Math. Softw., 22(3): 261–280, 1996.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    G.F. Jónsson and S. Vavasis. Solving polynomials with small leading coefficients. Siam Journal on Matrix Analysis and Applications, 26(2): 400–414, 2005.CrossRefGoogle Scholar
  22. [22]
    T. Lyche and J.M. Peña. Optimally stable multivariate bases. Advances in Computational Mathematics, 20: 149–159, January 2004.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    F. Tisseur and N.J. Higham. Structured pseudospectra for polynomial eigenvalue problems, with applications. SIAM J. Matrix Anal. Appl., 23(1): 187–208, 2001.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    V.Y. Pan. Coefficient-free adaptations of polynomial root-finders. Computers and Mathematics with Applications, 50: 263–369, 2005.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    N. Rezvani and R.M. Corless. The nearest polynomial with a given zero, revisited. Sigsam Bulletin, Communications on Computer Algebra, 134(3): 71–76, September 2005.MathSciNetGoogle Scholar
  26. [26]
    O. Ruatta. A multivariate Weierstrass iterative rootfinder. In ISSAC’ 01: Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, pages 276–283, New York, NY, USA, 2001. ACM Press.Google Scholar
  27. [27]
    A. Shakoori. The Bézout matrix in the Lagrange basis. In Laureano Gonzalez-Vega and Tomas Recio, editors, Proceedings EACA, pages 295–299, June 2004.Google Scholar
  28. [28]
    B.T. Smith. Error bounds for zeros of a polynomial based upon Gerschgorin’s theorem. Journal of the Association for Computing Machinery, 17(4): 661–674, October 1970.MATHMathSciNetGoogle Scholar
  29. [29]
    K.-C. Toh and L.N. Trefethen. Pseudozeros of polynomials and pseudospectra of companion matrices. Numerische Mathematik, 68: 403–425, 1994.MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    Y.-F. Tsai and R.T. Farouki. Algorithm 812: BPOLY: An object-oriented library of numerical algorithms for polynomials in Bernstein form. ACM Trans. Math. Softw., 27(2): 267–296, 2001.MATHCrossRefGoogle Scholar
  31. [31]
    J.R. Winkler. A comparison of the average case numerical condition of the power and Bernstein polynomial bases. Intern. J. Computer Math., 77: 583–602, 2001.MATHMathSciNetGoogle Scholar
  32. [32]
    J.R. Winkler. The transformation of the companion matrix resultant between the power and Bernstein polynomial bases. Appl. Numer. Math., 48(1): 113–126, 2004.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Robert M. Corless
    • 1
  1. 1.Ontario Research Centre for Computer Algebra and the Department of Applied MathematicsUniversity of Western OntarioLondonCanada

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