Values of minors of an infinite family of \(D\)-optimal designs and their application to the growth problem (Q2719199)

Suppose the matrix \(A=[a_{ij}] \in {\mathcal R}^{n \times n}\) is reduced to upper triangular form by using Gaussian elimination with complete pivoting (GECP). Let \(A^{(k)} = [a^{(k)}_{ij}]\) denote the matrix obtained after the first \(k\) pivoting operations, so that \(A^{(n-1)}\) is the final upper triangular matrix. A diagonal entry of that final matrix is called a pivot. Matrices with the property that no exchanges are actually needed during GECP are called completely pivoted (CP). Let \(g(n,A) =\) max\(_{i,j,k}|a^{(k)}_{ij}|/|a^{(0)}_{11}|\) denote the growth associated with GECP on \(A\) and \(g(n) =\) sup\(\{ g(n,A) \mid A \in {\mathcal R}^{n \times n}\}\). The problem of determining \(g(n)\) for various values of \(n\) is called the growth problem, and this remains one of the major unsolved problems in numerical analysis. One of the curious frustrations with this problem is that it is quite difficult to construct any examples of \(n \times n\) matrices \(A\) other than Hadamard matrices for which \(g(n,A)\) is even close to \(n\). \textit{J. Day} and \textit{B. Peterson} [Am. Math. Mon. 95, 489-513 (1988; Zbl 0654.65023)] have proved the equality \(g(n,A) = n\) for a certain class of \(n \times n\) Hadamard matrices. The authors of this paper in [Linear Algebra Appl. 306, 189-202 (2000; Zbl 0947.65031)] have also observed that weighing matrices of order \(n\) can give \(g(n,A) = n-1\). In this paper the authors obtain values for the pivots of 2-\(\{2s^2+2s+1; s^2; s^2; s(s-1)\}\) supplementary difference sets, and \(D\)-optimal designs made from them. Calculations have given moderate values of growth for \(D\)-optimal designs. Finally the authors mention an open problem concerning the possibility of finding \((1,-1)\) \(2v \times 2v\) CP \(D\)-optimal designs with growth greater than \(2v\).