Dynamic Data Structures in the Incremental Algorithms Operating on a Certain Class of Special Matrices

  • Jerzy S. Respondek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8584)


The article gives an incremental algorithm for calculating the inversion of the confluent Vandermonde matrix (CVM). We implemented all the incremental operations, i.e. adding, deleting and changing the single matrix parameter to avoid repeating the same calculations again and again. We obtained the quadratic logarithmic time complexity in the general case and the linear logarithmic complexity for the special case of small matrix parameters. Up to now, the fastest algorithms were of the cubic and quadratic class, respectively.


Binary Tree Data structures Numerical recipes Computational complexity 


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  1. 1.
    Adel’son-Vel’skii, G.M., Landis, E.M.: An algorithm for the organization of information. Soviet Mathematics Doklady 3, 1259–1263 (1962)Google Scholar
  2. 2.
    Appel, A.W.: An efficient program for many body simulation. SIAM J. Sci. Stat. Comput. 6(1), 85–103 (1985)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Augustyn, D.R., Warchal, L.: Cloud service solving N body problem based on Windows Azure platform. Comm. Comput. and Inf. Sci. 79, 84–95 (2010)CrossRefGoogle Scholar
  4. 4.
    Bentley, J.: Programming Pearls. AT & T Laboratories, New Jersey (1986)Google Scholar
  5. 5.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. McGraw Hill, Massachusetts (2001)zbMATHGoogle Scholar
  6. 6.
    Esselink, K.: The order of Appel’s algorithm. Inf. Proc. Lett. 41, 141–147 (1992)CrossRefzbMATHGoogle Scholar
  7. 7.
    Guibas, L.J., Sedgewick, R.: A dichromatic framework for balanced trees. In: 19th IEEE Annual Symposium on Foundations of Computer Science, pp. 8–21. IEEE Press (1978)Google Scholar
  8. 8.
    Hou, S., Pang, W.: Inversion of confluent Vandermonde matrices. Comput. and Math. with Appl. 43, 1539–1547 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Kincaid, D.R., Cheney, E.W.: Numerical Analysis: Mathematics of Scientific Computing, 3rd edn. Brooks Cole, California (2001)Google Scholar
  10. 10.
    Lee, K., O’Sullivan, M.E.: Algebraic soft-decision decoding of Hermitian codes. IEEE Trans. on Inf. Theory 56(6), 2587–2600 (2010)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C, 2nd edn. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  12. 12.
    Pugh, W.: Skip lists: A probabilistic alternative to balanced trees. Comm. of ACM 33(6), 668–676 (1990)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Respondek, J.S.: Approximate controllability of the n-th order infinite dimensional systems with controls delayed by the control devices. Int. J. Syst. Sci. 39(8), 765–782 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Respondek, J.S.: On the confluent Vandermonde matrix calculation algorithm. Appl. Math. Lett. 24, 103–106 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Respondek, J.S.: Numerical recipes for the high efficient inverse of the confluent Vandermonde matrices. Appl. Math. and Comp. 218, 2044–2054 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Respondek, J.S.: Recursive numerical recipes for the high efficient inverse of the confluent Vandermonde matrices. Appl. Math. and Comp. 225, 718–730 (2013)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Sakthivel, R., Ganesh, R., Anthoni, S.M.: Approximate controllability of fractional nonlinear differential inclusions. Appl. Math. and Comp. 225, 708–717 (2013)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Spitzbart, A.: A Generalization of Hermite’s interpolation formula. Am. Math. Mon. 67(1), 42–46 (1960)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Waite, W.M., Goos, G.: Compiler Construction, 2nd edn. Monographs in Computer Science. Springer, New York (1983)Google Scholar
  20. 20.
    Wiliams, J.W.: Algorithm 232 (Heapsort). Comm. of ACM 7, 347–348 (1964)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Jerzy S. Respondek
    • 1
  1. 1.Faculty of Automatic Control, Electronics and Computer Science, Institute of Computer ScienceSilesian University of TechnologyPoland

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