Abstract
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.
Preview
Unable to display preview. Download preview PDF.
Reference
Beth, T., Jungnickel, D., Lenz, H.: Design Theory. Cambridge University Press, Cambridge, Engalnd, 1986
Cohen, A. M.: A note on pivot size in Gaussian elimination. Lin. Alg. Appl. 8 (1974) 361–368
Cryer, C. W.: Pivot size in Gaussian elimination. Numer. Math. 12 (1968) 335–345
Day, J., Peterson, B.: Growth in Gaussian elimination. Amer. Math. Monthly 95 (1988) 489–513
Edelman, E., Mascarenhas, W.: On the complete pivoting conjecture for a Hadamard matrix of order 12. Linear and Multilinear Algebra 38 (1995) 181–187
Geramita, A. V., Seberry, J.: Orthogonal designs: Quadratic forms and Hadamard matrices. Marcel Dekker, New York-Basel, 1979
Gould, N.: On growth in Gaussian elimination with pivoting. SIAM J. Matrix Anal. Appl. 12 (1991) 354–361
Koukouvinos, C.: Linear models and D-optimal designs for n ≡ 2mod4. Statistics and Probability Letters 26 (1996) 329–332
Raghavarao, D.: Constructions and Combinatorial Problems in Design of Experiments. J. Wiley and Sons, New York, 1971
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) 431–560
Wilkinson, J. H.: The Algebraic Eigenvalue Problem. Oxford University Press, London, 1988
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mitrouli, M., Koukouvinos, C. (1997). On the growth problem for D-optimal designs. In: Vulkov, L., Waśniewski, J., Yalamov, P. (eds) Numerical Analysis and Its Applications. WNAA 1996. Lecture Notes in Computer Science, vol 1196. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62598-4_112
Download citation
DOI: https://doi.org/10.1007/3-540-62598-4_112
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-62598-8
Online ISBN: 978-3-540-68326-1
eBook Packages: Springer Book Archive