The symmetric orthogonalization, which is obtained from the polar decomposition of a matrix, is optimal. We propose an iterative algorithm to compute this orthogonalization on vector computers. It is especially efficient when the original matrix is near an orthonormal matrix.

In the well-known gossip problem, each of n gossips initially has a unique piece of information. The gossips can make a sequence of two-party telephone calls in which the two participants exchange every piece of information they have at the time of the call. The problem is to determine a minimum length sequence of telephone calls such that, by the end, everyone knows everyone else’s information. We consider Berman and Hawrylycz’s variation on this problem [this Journal, 7 (1986), pp. 13–17]. They introduce the additional feature that as many as k of the calls may fail in the sense that no information is exchanged, where k is a second parameter of the problem. We improve upon their upper bound on the minimum number of calls needed. This disproves a conjecture in the same paper. We also briefly consider the parallel complexity of this problem.

With a view to obtaining an efficient procedure for solving large-scale systems of equations, canonical block-triangular forms are defined for layered mixed matrices and for mixed matrices, and some practical examples are presented. The canonical forms are obtained from a straightforward application of the decomposition principle for submodular functions. The relation to the existing decomposition techniques for electrical networks, as well as to the Dulmage–Mendelsohn decomposition, is also discussed.

When a convection-diffusion equation is discretized using the central difference scheme the resulting coefficient matrix is not diagonally dominant whenever the convection terms are large. If this system of linear equations is solved using the conventional iteration methods, the iterations often fail to converge as some of the eigenvalues of the iteration matrix lie outside the unit circle $C = \{ z: | z |\leqq 1 \}$ in the complex plane. The eigenvalue spectrum of some of the iteration matrices lies inside the infinite strip $S = \{ z: | \operatorname{Real} ( z ) | < 1, | \operatorname{Imag} ( z ) | < \infty \}$. An example is that of the method of simultaneous displacements or the Jacobi method. In such cases, it is possible to enclose the eigenvalue spectrum inside an ellipse with major axis on the imaginary axis and minor axis in the real interval $( - 1,1 )$. This ellipse is used to define a convergent iteration. A practical computational algorithm is described to obtain such an iteration scheme. Numerical examples show that the spectrum enveloping technique works well when the original iterations diverge. When the original iterations converge the spectrum enveloping technique can converge even faster.

The Null Space Problem (NSP) is the following: Given a $t \times n$ matrix A with $t < n$, find a sparsest basis for its null space (a null basis). We show that columns in a sparsest null basis correspond to minimal dependent sets of columns of A. Sparsest null bases are characterized by a greedy algorithm that augments a partial basis by a sparsest null vector. Despite this result, (NSP) is NP-hard since finding a sparsest null vector of A is NP-complete. We prove that the related problem of finding a sparsest null basis with an embedded identity matrix is NP-hard too. Finally, we study the zero–nonzero structure of sparsest null bases.

The paper studies factorizations $a ( \lambda ) = b ( \lambda ) \cdot c ( \lambda )$ of $n \times n$ matrix polynomials, with emphasis on linear $b ( \lambda )$. It is assumed that $\det a ( \lambda )$ factorizes into scalar polynomials of degree 1. The tool is the concept of Keldysh chains (generalized Jordan chains). In Theorem 1 the linear independence of eigenvectors of a matrix is generalized to Keldysh chains.It is known that there is an $nm \times nm$ matrix polynomial $Kt ( m,a ( \lambda ) )$ such that the Keldysh chains of length $\leqq m$ with respect to an “eigenvalue” $\omega ( \det a ( \omega ) = 0 )$ may be identified with the nonzero vectors in the nullspace of $Kt ( m,a ( \omega ) )$. The main technique of the paper is to exploit the multiplicative property $Kt ( m,a ( \lambda ) ) = Kt ( m,b( \lambda ) ) \cdot Kt ( m,c ( \lambda ) )$ together with the mentioned generalization. Improved factorization results are obtained when det $a( \lambda )$ has more than $( n - 1 ) \cdot \deg a ( \lambd...

Given an $m \times n\, ( 0,1 )$ matrix A possessing special structure, we consider the Minimum Cost Maximal Covering Problem (MCP): \[ \operatorname{Min} \{ cx + dy|Ax + y\geqq 1^m ,1^n x\leqq p;x \in \{ 0,1 \}^n ,y \in \{ 0,1 \}^m \}. \] We give an $O ( p^2 n\min \{ m^2 ,n^2 \} )$ dynamic programming algorithm for solving (MCP) when the matrix A is totally balanced. Totally balanced matrices arise naturally in tree network location problems; however, the class of totally balanced matrices is not equivalent to the class of covering matrices arising from tree network location problems: we give an example of a totally balanced matrix for which there exists no corresponding tree network covering problem.

A short proof is given for the theorem that in a rectilinear, simply connected art gallery, $[ n /4 ]$ watchmen are sufficient, where n is the number of the corners, in order for at least one to have a view of each internal point.

In this paper bounds are developed and investigated on the growth factors and the multipliers resulting from Gaussian elimination applied to an irreducible tridiagonal H-matrix. These results extend the study of the stability of Gaussian elimination without pivoting on certain tridiagonal matrices by Gunzburger and Nicolaides.

Previous article Full AccessErratum: Volterra Multipliers IIRay RedhefferRay Redhefferhttps://doi.org/10.1137/0607037PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"Erratum: Volterra Multipliers II." SIAM Journal on Algebraic Discrete Methods, 7(2), p. 336 Previous article FiguresRelatedReferencesCited byDetails Volume 7, Issue 2| 1986SIAM Journal on Algebraic Discrete Methods History Published online:17 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0607037Article page range:pp. 336-336ISSN (print):0196-5212ISSN (online):2168-345XPublisher:Society for Industrial and Applied Mathematics