Some Hypermatrix Algorithms in Linear Algebra

  • K. A. Braun
  • G. Dietrich
  • G. Frik
  • Th. L. Johnsen
  • K. Straub
  • G. Vallianos
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 134)


The efficient solution of large linear matrix problems plays a central role in both linear and nonlinear structural analysis. Accordingly, a substantial effort has been allocated for the design of computer software in order to handle standard tasks like the solution of linear equations or eigenreduction.


Finite Element Technique Jacobi Method Operation Count Inverse Iteration Hessenberg Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J.H. Wilkinson and C. Reinsch, Handbook for automatic computation, II, Linear Algebra (Springer Verlag, 1971 ).Google Scholar
  2. [2]
    J.H. Wilkinson, The algebraic eigenvalue problem ( Clarendon Press, Oxford, 1965 ).MATHGoogle Scholar
  3. [3]
    J.G.F. Francis, The QR-transformation. A unitary analogue to the LR-transformation. Part 1: Computer Journal 4 (1961) 265–271; Part 2: Computer Journal 4 (1962) 332–345.MathSciNetGoogle Scholar
  4. [4]
    A.C. McKellar and E.G. Coffman, Organizing matrices and matrix operations for paged memory systems, Comm. of A.C.M. 12, 3 (1969) 153–165.MATHGoogle Scholar
  5. [5]
    G. von Fuchs, J.R. Roy and E. Schrem, “Hypermatrix solution of large sets of symmetric positive-definite linear equations, Comp. Meth. Appl. Mech. Eng. 1 (1972) 197–216.MATHCrossRefGoogle Scholar
  6. [6]
    G. von Fuchs and E. Schrem, “ASKA - A computer system for structural engineers”, Proceedings of the I.S.S.C. Symposium on finite element techniques, ISD, University of Stuttgart (1969).Google Scholar
  7. [7]
    J.H. Argyris, “Continua and Discontinua”, Opening address to the international conference on matrix methods of structural mechanics, Dayton, Ohio. Wright-Patterson U.S.A.F. Base, October 26th, 1965, published in the Proceedings of the Conference by U.S. Government (1967) 1–198.Google Scholar
  8. [8]
    G.M. Skagestein, “Rekursiv unterteilte Matrizen und Methoden zur Erstellung von Rechenprogrammen für ihre Verarbeitung”, Dr. Ing. Thesis, University of Stuttgart, 1971.Google Scholar
  9. [9]
    O.E. Brönlund and Th. Lunde Johnsen, QR-factorization of partitioned matrices, Comp.Meth.Appl.Mech.Eng. 3 (1974) 153–172.MATHCrossRefGoogle Scholar
  10. [10]
    J.H. Argyris and O.E.Brönlund, The natural factor formulation of the matrix displacement method, Comp. Meth. Appl. Mech. Eng. 5 (1975) 97–119.MATHGoogle Scholar
  11. [11]
    Th. Lunde Johnsen and J.R. Roy, On system., of linear equations of the form AtAx = b error analysis and certain consequences for structural applications, Comp.Meth.Appl.Mech. Eng. 3 (1974) 357–374.MATHGoogle Scholar
  12. [12]
    J.W. Backus et al, Revised report on the algorithmic language ALGOL 60, IFIP 1962.Google Scholar
  13. [13]
    O.E. Brönlund, “Eigenvalues of large matrices”, Symposium on finite element techniques at the ISD, University of Stuttgart (1969).Google Scholar
  14. [14]
    W.M. Gentleman, Least squares computation by Givens transformations without square roots, J. Inst.Maths.Applics., 12 (1973) 329–336.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    P.J. Eberlein, A Jacobi-like method for the automatic computation of eigenvalues and eigen-vectors of an arbitrary matrix, J. Soc. Indust.Appl.Math. 10 (1962) 74–88.MathSciNetCrossRefGoogle Scholar
  16. [16]
    K.A. Braun, Th.Lunde Johnsen, “Hypermatrix generalisation of the Jacobi and EberleinMethod for computing eigenvalues and eigenvectors of Hermitean and non-Hermitean matrices”, Comp.Meth.Appl.Mech.Eng. 4 (1974) 1–18.MATHCrossRefGoogle Scholar
  17. [17]
    H. Rutishauser, Computational aspects of F.J. Bauer’s simultaneous iteration method, Numerische Mathematik 13 (1969) 4–13.MATHCrossRefGoogle Scholar
  18. [18]
    P.S. Jensen, The solution of large symmetric eigenproblems by sectioning, SIAM J. Numer. Anal. 9 (1972) 534–545.MATHCrossRefGoogle Scholar
  19. [19]
    S. Falk, Berechnung von Eigenwerten und Eigenvektoren normaler Matrizenpaare durch Ritz-Iteration, ZAMM 53 (1973) 73–91.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    H.R. Schwarz, The eigenvalue problem (A AB)x=0 for symmetric matrices of high order, Comp.Meth.Appl.Mech.Eng. 3 (1974) 11–28.MATHCrossRefGoogle Scholar
  21. [21]
    C.C. Paige and M.A. Saunders, Solution of sparse indefinite systems of linear equations, SIAM J. Numer.Anal. 12 (1975) 617–629.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    K.A. Braun and Th. Lunde Johnsen, “Eigencomputation of symmetric hypermatrices using a generalization of the Householder method”, Lecture given at the 2nd International Conference on Structural Mechanics in Reactor Technology, Berlin, 10–14 Sept. 1973.Google Scholar
  23. [23]
    UM 212 ASKA Part II, Linear dynamic analysis, Lecture Notes and example problems. ISD-Report No. 155, University of Stuttgart, 1974.Google Scholar
  24. [24]
    G. Dietrich, “A new formulation of the Hyper-QR-decomposition and related algorithms”, Lecture given at the 3rd Post Conference on Computational Aspects of the Finite Element Method, Imperial College, London, 8–9 September 1975.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1976

Authors and Affiliations

  • K. A. Braun
    • 1
  • G. Dietrich
    • 1
  • G. Frik
    • 1
  • Th. L. Johnsen
    • 1
  • K. Straub
    • 1
  • G. Vallianos
    • 1
  1. 1.Institut für Statik und Dynamik der Luft- und RaumfahrtkonstruktionenUniversity of StuttgartGermany

Personalised recommendations