Numerical Algorithms

, Volume 42, Issue 3–4, pp 345–361 | Cite as

Interpolation algorithm of Leverrier–Faddev type for polynomial matrices



We investigated an interpolation algorithm for computing outer inverses of a given polynomial matrix, based on the Leverrier–Faddeev method. This algorithm is a continuation of the finite algorithm for computing generalized inverses of a given polynomial matrix, introduced in [11]. Also, a method for estimating the degrees of polynomial matrices arising from the Leverrier–Faddeev algorithm is given as the improvement of the interpolation algorithm. Based on similar idea, we introduced methods for computing rank and index of polynomial matrix. All algorithms are implemented in the symbolic programming language MATHEMATICA , and tested on several different classes of test examples.


pseudoinverse interpolation MATHEMATICA Leverrier–Faddeev method, polynomial matrices 

Mathematics Subject Classifications (2000)

15A09 69Q40 


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  1. 1.
    Ben-Israel, A., Greville, T.N.E.: Generalized Inverses. Theory and Applications, 2nd edn, Ouvrages de Mathmatiques de la SMC, 15. Springer, Berlin Heidelberg New York (2003)Google Scholar
  2. 2.
    Decell, H.P.: An application of the Cayley–Hamilton theorem to generalized matrix inversion. SIAM Rev. 7(4), 526–528 (1965)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Ji, J.: A finite algorithm for the Drazin inverse of a polynomial matrix. Appl. Math. Comput. 30, 243–251 (2002)CrossRefGoogle Scholar
  4. 4.
    Ji, J.: Explicit expressions of the generalized inverses and condensed Cramer rules. Linear Algebra Appl. 404, 183–192 (2005)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Jones, J., Karampetakis, N.P., Pugh, A.C.: The computation and application of the generalized inverse vai Maple. J. Symb. Comput. 25, 99–124 (1998)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Karampetakis, N.P.: Computation of the generalized inverse of a polynomial matrix and applications. Linear Algebra Appl. 252, 35–60 (1997)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Karampetakis, N.P.: Generalized inverses of two-variable polynomial matrices and applications. Circuits Syst. Signal Process. 16, 439–453 (1997)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Karampetakis, N.P., Tzekis, P.: On the computation of the generalized inverse of a polynomial matrix. IMA J. Math. Control Inf. 18, 83–97 (2001)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Li, X., Wei, Y.: A note on computing the generalized inverse A T,S (2) of a matrix A. Int. J. Math. Math. Sci. 31, 497–507 (2002)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Stanimirovic, P.S., Tasić, M.B.: Drazin inverse of one-variable polynomial matrices. Filomat, Niš, 15, 71–78 (2001)Google Scholar
  11. 11.
    Stanimirović, P.S.: A finite algorithm for generalized inverses of polynomial and rational matrices. Appl. Math. Comput. 144, 199–214 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Press, W.H., Teukolsky, S.A., Wetterling, W.T., Flannery, B.P.: Numerical Receipts in C. Cambridge University Press, UK (1992)Google Scholar
  13. 13.
    Schuster, A., Hippe, P.: Inversion of polynomial matrices by interpolation. IEEE Trans. Automat. Contr. 37(3), 363–365 (1992)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wang, G., Wei, Y., Qiao, S.: Generalized Inverses: Theory and Computations Science, Beijing, China (2004)Google Scholar
  15. 15.
    Bu, F., Wei, Y.: The algorithm for computing the Drazin inverse of two-variable polynomial matrices. Appl. Math. Comput. 147, 805–836 (2004)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Wei, Y.: A characterization and representation of the generalized inverse A T,S (2) and its applications. Linear Algebra Appl. 280, 87–96 (1998)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Wei, Y., Djordjević, D.S.: On integral representation of the generalized inverse A T,S (2). Appl. Math. Comput. 142, 189–194 (2003)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Wei, Y., Wu, H.: The representation and approximation for the generalized inverse A T,S (2). Appl. Math. Comput. 135, 263–276 (2003)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Wei, Y., Zhang, N.: A note on the representation and approximation of the outer inverse A T,S (2) of a matrix A. Appl. Math. Comput. 147(3), 837–841 (2004)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Wolfram, S.: The Mathematica Book, 4th edn., Cambridge University Press, UK (1999)MATHGoogle Scholar
  21. 21.
    Zielke, G.: Report on test matrices for generalized inverses. Computing 36, 105–162 (1986)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  • Marko D. Petković
    • 1
  • Predrag S. Stanimirović
    • 1
  1. 1.Department of Mathematics, Faculty of ScienceUniversity of NišNišSerbia and Montenegro

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