SUMMARYThe direct methods for the solution of systems of linear equations with a symmetric positive-semidefinite (SPS) matrix A usually comprise the Cholesky decomposition of a nonsingular diagonal block A JJ of A and effective evaluation of the action of a generalized inverse of the corresponding Schur complement. In this note we deal with both problems, paying special attention to the stiffness matrices of floating structures without mechanisms. We present a procedure which first identifies a well-conditioned positive-definite diagonal block A JJ of A, then decomposes A JJ by the Cholesky decomposition, and finally evaluates a generalized inverse of the Schur complement S of A JJ . The Schur complement S is typically very small, so the generalized inverse can be effectively evaluated by the singular value decomposition (SVD). If the rank of A or a lower bound on the nonzero eigenvalues of A are known, then the SVD can be implemented without any 'epsilon'. Moreover, if the kernel of A is known, then the SVD can be replaced by effective regularization. The results of numerical experiments show that the proposed method is useful for effective implementation of the FETI-based domain decomposition methods.

It was proved by Fronček, Jerebic, Klavžar, and Kovář that if a complete bipartite graph K n,n with a perfect matching removed can be covered by k bicliques, then n k k 2 . We give a slightly simplified proof and we show that the result is tight. Moreover, we use the result to prove analogous bounds for coverings of some other classes of graphs by bicliques.

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The strong isometric dimension of a graph G is the least number k such that G isometrically embeds into the strong product of k paths. Using Sperner's theorem, the strong isometric dimension of the Hamming graphs K 2 K n is determined.

We propose a method of a parallel distribution of densely populated matrices arising in boundary element discretizations of partial differential equations. In our method the underlying boundary element mesh consisting of n elements is decomposed into N submeshes. The related N ×N submatrices are assigned to N concurrent processes to be assembled. Additionally we require each process to hold exactly one diagonal submatrix, since its assembling is typically most time consuming when applying fast boundary elements. We obtain a class of such optimal parallel distributions of the submeshes and corresponding submatrices by cyclic decompositions of undirected complete graphs. It results in a method the theoretical complexity of which is O((n/ √ N) log(n/ √ N)) in terms of time for the setup, assembling, matrix action, as well as memory consumption per process. Nevertheless, numerical experiments up to n = 2744832 and N = 273 on a real-world geometry document that the method exhibits superior parallel scalability O((n/N) log n) of the overall time, while the memory consumption scales accordingly to the theoretical estimate.