, Volume 52, Issue 2, pp 203–211 | Cite as

A fast algorithm for solving special tridiagonal systems

  • W. -M. Yan
  • K. -L. Chung
Short Communication


In this paper, a fast algorithm for solving the special tridiagonal system is presented. This special tridiagonal system is a symmetric diagonally dominant and Toeplitz system of linear equations. The error analysis is also given. Our algorithm is quite competitive with the Gaussian elimination, cyclic reduction, specialLU factorization, reversed triangular factorization, and Toeplitz factorization methods. In addition, our result can be applied to solve the near-Toeplitz tridiagonal system. Some examples demonstrate the good efficiency and stability of our algorithm.

AMS Subject Classification


Key words

Diagonally dominant matrices error analysis linear recurrences Toeplitz matrices tridiagonal matrices 

Ein schneller Algorithmus zur Lösung spezieller tridiagonaler Systeme


In dieser Arbeit wird ein schneller Algorithmus zur Lösung symmetrischer, diagonaldominanter tridiagonaler Töpflitz-Systeme vorgestellt. Auch eine Fehleranalyse liegt vor. Der Algorithmus ist den folgenden Verfahren mindestens gleichwertig: Gauss-Elimination, zyklische Reduktion, spezielleLU-Faktorisierung, umgekehrte Faktorisierung, Töplitz-Faktorisierung. Außerdem kann unser Vorgehen zur Lösung in tridiagonalen fast-Töplitz-Systemen verwendet werden. Einige Beispiele zeigen die Effizienz und Stabilität unseres Algorithmus.


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  1. [1]
    Barlett, M. S.: An inverse matrix adjustment arising in discriminant analysis. Ann. Math. Statist.22, 107–111 (1951).Google Scholar
  2. [2]
    Boisvert, R. F.: Algorithms for special tridiagonal systems. SIAM J. Sci. Stat. Comput.12, 423–442 (1991).Google Scholar
  3. [3]
    Chen, M. K.: On the solution of circulant linear systems. SIAM J. Numer. Anal.24, 668–683 (1987).Google Scholar
  4. [4]
    Dongarra, J. J., Moler, C. B., Bunch, J. R., Stewart, G. W.: LINPACK user's guide. SIAM Press, 1979.Google Scholar
  5. [5]
    Evans, D. J., Forrington, C. V. D.: Note on the solution of certain tri-diagonal systems of linear equations. Comput. J.5, 327–328 (1963).Google Scholar
  6. [6]
    Evans, D. J.: An algorithm for the solution of certain tri-diagonal systems of linear equations. Comput. J.15, 356–359 (1972).Google Scholar
  7. [7]
    Evans, D. J.: On the solution of certain Toeplitz tridiagonal linear systems. SIAM J. Numer. Anal.17, 675–680 (1980).Google Scholar
  8. [8]
    Fischer, D., Golub, G., Hald, O., Levia, C., Winlund, O.: On Fourier-Toeplitz methods for separable elliptic problems. Math. Comput.28, 349–368 (1974).Google Scholar
  9. [9]
    Hockney, R. W.: A fast direct solution of Poisson's equation using Fourier analysis. J. ACM.12, 95–113 (1965).Google Scholar
  10. [10]
    Malcolm, M. A., Palmer, J.: A fast method for solving a class of tridiagonal linear systems. Comm. ACM.17, 14–17 (1974).Google Scholar
  11. [11]
    Pham, B.: Quadratic B-splines for automatic curve and surface fitting. Comput. Graphics13, 471–475 (1989).Google Scholar
  12. [12]
    Rojo, O.: A new method for solving symmetric circulant tridiagonal systems of linear equations. Comput. Math. Appl.20, 61–67 (1990).Google Scholar
  13. [13]
    Smith, G. D.: Numerical solution of partial differential equations: finite difference methods, 3rd edn. New York: Oxford University Press 1985.Google Scholar
  14. [14]
    Widlund, O. B.: On the use of fast methods for separable finite difference equations for the solution of general elliptic problems. In: Sparse matrices and their applications (Rose, D. J., Willoughby, R. A., eds.), pp. 121–131. New York: Plenum Press 1972.Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • W. -M. Yan
    • 1
  • K. -L. Chung
    • 2
  1. 1.Department of Computer Science and Information EngineeringNational Taiwan UniversityTaipei, TaiwanR.O.C.
  2. 2.Department of Information ManagementNational Taiwan Institut of TechnologyTaipei, TaiwanR.O.C.

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