A new orthogonalization procedure with an extremal property

A new orthogonalization procedure with an extremal property

Various methods of constructing an orthonomal set out of a given set of linearly independent vectors are discussed. Particular attention is paid to the Gram-Schmidt and the Schweinler-Wigner orthogonalization procedures. A new orthogonalization procedure which, like the SchweinlerWigner procedure, is democratic and is endowed with an extremal property is suggested. PACS No: 02.10.Sp, 31.15.+q e...

A Block Orthogonalization Procedure with Constant Synchronization Requirements

We propose an alternative orthonormalization method that computes the orthonor-mal basis from the right singular vectors of a matrix. Its advantage are: a) all operations are matrix-matrix multiplications and thus cache-eecient, b) only one synchronization point is required in parallel implementations, c) could be more stable than Gram-Schmidt. In addition, we consider the problem of incrementa...

An Extremal Property of Fekete Polynomials

An Extremal Property of Turán Graphs

Let Fn,tr(n) denote the family of all graphs on n vertices and tr(n) edges, where tr(n) is the number of edges in the Turán’s graph Tr(n) – the complete r-partite graph on n vertices with partition sizes as equal as possible. For a graph G and a positive integer λ, let PG(λ) denote the number of proper vertex colorings of G with at most λ colors, and let f(n, tr(n), λ) = max{PG(λ) : G ∈ Fn,tr(n...

An extremal property of Bernstein operators

We establish a strong version of a known extremal property of Bernstein operators, as well as several characterizations of a related specific class of positive polynomial operators. © 2006 Elsevier Inc. All rights reserved. MSC: 41A36; 26D15