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Choleski-Banachiewicz Approach to Systems with Non-positive Definite Matrices with Mathematica®

  • Ryszard A. Walentyński
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3039)

Abstract

The paper presents the discussion on extension of potential application of the Choleski-Banachiewicz algorithm to the system of linear equations with non-positive definite matrices. It is shown that the method is also stable in case of systems with negative definite matrices and sometimes may be also successful if the matrix is neither positive nor negative definite. The algorithm handles systems with complex symmetric (not Hermitian) matrices, too. This fact has deep physical sense and engineering applications since systems with negative definite matrices are common in tasks of dynamics and post buckling analysis in civil and mechanical engineering. Possibility of utilization of Choleski-Banachiewicz algorithm to such problems can be very practical. The entire analysis has been carried out within Mathematica ® environment.

References

  1. 1.
    Banachiewicz, T.: Mcthode dc Resolution Numerique des Equations Lineaires, Du Calcul des Determinants et des Inverses et de Reduction des Formes Quadratiques. Bull. Ac. Pol. Sci. A, 393-404 (1938)Google Scholar
  2. 2.
    Bcnoit, A.: Note Surface ton Methode, Proce de Du Commendenl Cholesky. Bull. Geod. Toulouse, 67-77 (1924)Google Scholar
  3. 3.
    Butatovic, R.M.: On Stability Criteria for Gyroscopic Systems with Negative Definite Stiffness. Facta Universitatis, series Mcchanica, Automatic Control and Robotics 2(10), 1081–1087 (2000), http://facta.junis.ni.ac.yu/facta/macar/macar2000/macar2000-07.pdf Google Scholar
  4. 4.
    Burden, R.L., Faires, J.D.: Numerical Analysis, 5th edn. PWS Publishing Company, Boston (1993)zbMATHGoogle Scholar
  5. 5.
    Dolittlc, M.H.: Method Employed in the Solution of Normal Equations and the Adjustment of Transquation. U. S. Survey Report, 115-120 (1878)Google Scholar
  6. 6.
    Janczura, A.: Solving of Linear Equations with Bounded Right-Hand Side. Mcchanika i Komputer, Warsaw 8, 235–250 (1990)zbMATHGoogle Scholar
  7. 7.
    Langer, J.: Dynamics of Bar Structures. In: Structural Mechanics with Elements of Computer Approach. ch. 7, vol. 2, pp. 76–156. Arkady, Warsaw (1984)Google Scholar
  8. 8.
    Paltison, D.H.: Summary: Pronunciation of ’Cholesky’. S-Ncws Mailing List Archives Division of Biostatistics at Washington University School of Medicine, Saint Louis (2000) http://www.biostat.wustl.edu/archivcs/html/s-news/2000-10/msg00172.html
  9. 9.
    Walentynski, R.A.: Refined Least Squares Method for Shells Boundary-Value Problems. In: Tazawa, Y., et al. (eds.) Symbolic Computation: New Horizons. Proceedings of the Fourth international MATHMATUA Symposium, pp. 511–518. Tokyo Denki University Press, Tokyo (2001) (extended version on the proceedings CD-ROM)Google Scholar
  10. 10.
    Walentynski, R.A.: Application of Computer Algebra in Symbolic Computations and Boundary-Value Problems of the Theory of Shells. Silesian University of Technology Press, Gliwice, Civil Engineering 1587(100) (2003)Google Scholar
  11. 11.
    Weisstein, E.: Cholesky Decomposition. In: World of Mathematics, a Wolfram Web Resource, CRC Press LLC, Wolfram Research, Inc., Boca Raton Champaign (2003), http://mathworld.wolfram.com/CholeskyDecomposition.html Google Scholar
  12. 12.
    Research, W.: CholcskyDecomposition. In: Built-in Functions, chapter Lists and Matrices - Matrix Operations, Wolfram Media, Champaign (2003), http://documents.wolfram.com/v5/Built-inFunctions/ListsAndMatrices/MatrixOperations/CholeskyDecomposition.html Google Scholar
  13. 13.
    Wolfram Research: MatrixManipulation. In: Standard Add-on Packages, chapter Linear Algebra. Wolfram Media, Champaign (2003), http://documcnts.wolfram.com/v5/Add-onsLinks/StandardPackages/LinearAlgebra/MatrixManipulation.html
  14. 14.
    Wolfram, S.: Advanced Matrix Operations. In: The MATHEMATICA® Book - Online, 5th edn., ch. 3.7.10, Wolfram Media, Champaign (2003), http://documents.wolfram.com/v5/TheMathematicaBook/AdvancedMathematicsInMathematica/LinearAlgebra/3.7.10.html Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Ryszard A. Walentyński
    • 1
  1. 1.Faculty of Civil EngineeringSilesian University of TechnologyGliwicePoland

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