Christos Kravvaritis

The growth factor of a Hadamard matrix of order 16 is 16

In 1968 Cryer conjectured that the growth factor of an n × n Hadamard matrix is n. In 1988 Day and Peterson proved this only for the Hadamard–Sylvester class. In 1995 Edelman and Mascarenhas proved that the growth factor of a Hadamard matrix of order 12 is 12. In the present paper we demonstrate the pivot structures of a Hadamard matrix of order 16 and prove for the first time that its growth factor is 16. The study is divided in two parts: we calculate pivots from the beginning and pivots from the end of the pivot pattern. For the first part we develop counting techniques based on symbolic manipulation for specifying the existence or non-existence of specific submatrices inside the first rows of a Hadamard matrix, and so we can calculate values of principal minors. For the second part we exploit sophisticated numerical techniques that facilitate the computations of all possible (n − j) × (n − j) minors of Hadamard matrices for various values of j. The pivot patterns are obtained by utilizing appropriately the fact that the pivots appearing after the application of Gaussian elimination on a completely pivoted matrix are given as quotients of principal minors of the matrix. Copyright

Compact Fourier Analysis for Multigrid Methods based on Block Symbols

The notion of compact Fourier analysis (CFA) is discussed. CFA allows description of multigrid (MG) in a nutshell and offers a clear overview on all MG components. The principal idea of CFA is to model the MG mechanisms by means of scalar generating functions and matrix functions (block symbols). The formalism of the CFA approach is presented by describing the symbols of the fine and coarse grid problems, the prolongation and restriction, the smoother, and the coarse grid correction, resp., smoothing corrections. CFA uses matrix functions and their features (e.g., product, inverse, adjugate, norm, spectral radius, eigenvectors, eigenvalues of multilevel

On the pivot structure for the weighing matrix W(12, 11)

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On the complete pivoting conjecture for Hadamard matrices: further progress and a good pivots property

-circulant matrices), and scalar functions and their roots. This leads to an elementary description and allows for an easy analysis of MG algorithms. A first application is to utilize CFA for deriving MG as a direct solver, i.e., an MG cycle that will converge in just one iteration step. Necessary and sufficient conditions that have to be fulfilled by the MG components are given for obtaining MG functioning as a direct solver. Furthermore, new general and practical smoothers and transfer operators that lead to efficient MG methods are introduced. In addition, we study sparse approximations of the Galerkin coarse grid operator yielding efficient and practicable MG algorithms (approximately direct solvers). Numerical experiments demonstrate the theoretical results.

Properties of submatrices of Sylvester Hadamard matrices

In the present article we concentrate our study on the growth problem for the weighing matrix W(12,11) and show that the unique W(12,11) has three pivot structures. An improved algorithm for extending a k × k (0,+,-) matrix to a W(n,n-1), if possible, has been developed to simplify the proof. For the implementation of the algorithm special emphasis is given to the notions of data structures and parallel processing.

Compound matrices: properties, numerical issues and analytical computations

Further progress is achieved for the growth conjecture for Hadamard matrices. It is proved that the leading principal minors of a CP Hadamard matrix form an increasing sequence. Bounds for the sixth and seventh pivot of any CP Hadamard matrix are given. A new proof demonstrating that the growth of a Hadamard matrix of order 12 is 12, is presented. Moreover, a new notion of good pivots is introduced and its importance for the study of the growth problem for CP Hadamard matrices is examined. We establish that CP Hadamard matrices with good pivots satisfy Cryer’s growth conjecture with equality, namely their growth factor is equal to their order. A construction of an infinite class of Hadamard matrices is proposed.

Modified Jacobian Newton Iterative Method with Embedded Domain Decomposition Method

Abstract We investigate the algebraic behaviour of leading principal submatrices of Hadamard matrices being powers of 2. We provide analytically the spectrum of general submatrices of these Hadamard matrices. Symmetry properties and relationships between the upper left and lower right corners of the matrices in this respect are demonstrated. Considering the specific construction scheme of this particular class of Hadamard matrices (called Sylvester Hadamard matrices), we utilize tensor operations to prove the respective results. An algorithmic procedure yielding the complete spectrum of leading principal submatrices of Sylvester Hadamard matrices is proposed.

An algorithm to find values of minors of skew hadamard and conference matrices

This paper studies the possibility to calculate efficiently compounds of real matrices which have a special form or structure. The usefulness of such an effort lies in the fact that the computation of compound matrices, which is generally noneffective due to its high complexity, is encountered in several applications. A new approach for computing the Singular Value Decompositions (SVD’s) of the compounds of a matrix is proposed by establishing the equality (up to a permutation) between the compounds of the SVD of a matrix and the SVD’s of the compounds of the matrix. The superiority of the new idea over the standard method is demonstrated. Similar approaches with some limitations can be adopted for other matrix factorizations, too. Furthermore, formulas for the n − 1 compounds of Hadamard matrices are derived, which dodge the strenuous computations of the respective numerous large determinants. Finally, a combinatorial counting technique for finding the compounds of diagonal matrices is illustrated.

Computations for Minors of Weighing Matrices with Application to the Growth Problem

In this article a new approach is proposed for constructing a domain decomposition method based on the iterative operator-splitting method for nonlinear differential equations. The convergence properties of such a method are studied. The main feature of the proposed idea are the linearization of the nonlinear equations and the application of iterative splitting methods. We present iterative operator-splitting method with embedded Newton methods to solve the nonlinearity. We confirm with numerical applications the effectiveness of the proposed iterative operator-splitting method in comparison with the classical Newton methods. We provide improved results and convergence rates.

Hadamard Matrices: Insights into Their Growth Factor and Determinant Computations

We give an algorithm to obtain formulae and values for minors of skew Hadamard and conference matrices. One step in our algorithm allows the (n–j) × (n–j) minors of skew Hadamard and conference matrices to be given in terms of the minors of a 2j−1 × 2j−1 matrix. In particular we illustrate our algorithm by finding explicitly all the (n-3) × (n-3) minors of such matrices.