On the growth problem for D-optimal designs

  • M. Mitrouli
  • C. Koukouvinos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1196)


When Gaussian elimination with complete pivoting (GECP) is applied to a real n×n matrix A, we will call g(n, A) the associated growth of the matrix. The problem of determining the largest growth g(n) for various values of n is called the growth problem. It seems quite difficult to establish a. value or close bounds for g(n). For specific values of n (n=1, 2, 3, 4) and for a special category of matrices, such as Hadamard matrices, g(n) has been evaluated exactly. In the present paper, we discuss the maximum determinant and the growth problem of n×n matrices with elements ±1, which are called D-optimal designs. Specific examples of n×n weighing matrices W attaining g(n, W)=n −1 are exhibited.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beth, T., Jungnickel, D., Lenz, H.: Design Theory. Cambridge University Press, Cambridge, Engalnd, 1986Google Scholar
  2. 2.
    Cohen, A. M.: A note on pivot size in Gaussian elimination. Lin. Alg. Appl. 8 (1974) 361–368Google Scholar
  3. 3.
    Cryer, C. W.: Pivot size in Gaussian elimination. Numer. Math. 12 (1968) 335–345Google Scholar
  4. 4.
    Day, J., Peterson, B.: Growth in Gaussian elimination. Amer. Math. Monthly 95 (1988) 489–513Google Scholar
  5. 5.
    Edelman, E., Mascarenhas, W.: On the complete pivoting conjecture for a Hadamard matrix of order 12. Linear and Multilinear Algebra 38 (1995) 181–187Google Scholar
  6. 6.
    Geramita, A. V., Seberry, J.: Orthogonal designs: Quadratic forms and Hadamard matrices. Marcel Dekker, New York-Basel, 1979Google Scholar
  7. 7.
    Gould, N.: On growth in Gaussian elimination with pivoting. SIAM J. Matrix Anal. Appl. 12 (1991) 354–361Google Scholar
  8. 8.
    Koukouvinos, C.: Linear models and D-optimal designs for n ≡ 2mod4. Statistics and Probability Letters 26 (1996) 329–332Google Scholar
  9. 9.
    Raghavarao, D.: Constructions and Combinatorial Problems in Design of Experiments. J. Wiley and Sons, New York, 1971Google Scholar
  10. 10.
    Seberry, J., Yamada, M.: Hadamard matrices, sequences and block designs. Contemporary Design Theory: A collection of surveys. Edited by J. Dinitz and D. R. Stinson, J. Wiley and Sons, New York, (1992) 431–560Google Scholar
  11. 11.
    Wilkinson, J. H.: The Algebraic Eigenvalue Problem. Oxford University Press, London, 1988Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • M. Mitrouli
    • 1
  • C. Koukouvinos
    • 2
  1. 1.Department of MathematicsUniversity of AthensAthensGreece
  2. 2.Department of MathematicsNational Technical University of AthensAthensGreece

Personalised recommendations