Christos Kravvaritis

This paper discusses templates for designing efficient and robust multigrid (MG) methods. All MG mechanisms are described by means of matrix functions (block symbols), which capture the properties of the full matrices. We propose a new strategy for deriving the block symbol by means of sparse matrices and of its representation in terms of the usual tensor product. Emphasis is laid on the indefinite Helmholtz equation, which is of great practical interest in real‐life applications. We take advantage of previously known patterns (the subblock smoother as preconditioner and the full projection as grid transfer operator) for designing MG methods that work as direct solvers. The corresponding numerical results are presented and the superiority of the combined use of a subblock smoother and a full projection is demonstrated. The condition number and positive definiteness of the subblock smoother especially for Helmholtz‐like problems are analyzed, and the corresponding requirements and bounds are given. Copyright © 2013 John Wiley & Sons, Ltd.

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 © 2009 John Wiley & Sons, Ltd.