Bounds for the Condition Number of Polynomials Systems with Integer Coefficients

Suppose ai indicates the number of orbits of size i in graph G. A new counting polynomial, namely an orbit polynomial, is defined as OG(x) = ∑i aixi. Its modified version is obtained by subtracting the orbit polynomial from 1. In the present paper, we studied the conditions under which an integer polynomial can arise as an orbit polynomial of a graph. Additionally, we surveyed graphs with a small number of orbits and characterized several classes of graphs with respect to their orbit polynomials.

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Some Characterizations of the Distribution of the Condition Number of a Complex Gaussian Matrix

Special Matrices ◽

10.1515/spma-2020-0004 ◽

2020 ◽

Vol 8 (1) ◽

pp. 22-35

Author(s):

M. Shakil ◽

M. Ahsanullah

Keyword(s):

Order Statistics ◽

Condition Number ◽

Record Values ◽

Distributional Properties ◽

Upper Record Values ◽

Upper Record

AbstractThe objective of this paper is to characterize the distribution of the condition number of a complex Gaussian matrix. Several new distributional properties of the distribution of the condition number of a complex Gaussian matrix are given. Based on such distributional properties, some characterizations of the distribution are given by truncated moment, order statistics and upper record values.

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Schur complement spectral bounds for large hybrid FETI-DP clusters and huge three-dimensional scalar problems

Journal of Numerical Mathematics ◽

10.1515/jnma-2020-0048 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Zdeněk Dostál ◽

Tomáš Brzobohatý ◽

Oldřich Vlach

Keyword(s):

Condition Number ◽

Three Dimensional ◽

Schur Complement ◽

Decomposition Methods ◽

Stiffness Matrices ◽

Spectral Bounds ◽

Schur Complements ◽

Linear Problems ◽

Primal Method ◽

Large Clusters

Abstract Bounds on the spectrum of Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients of the convergence analysis of FETI (finite element tearing and interconnecting) based domain decomposition methods. Here we give bounds on the regular condition number of Schur complements of “floating” clusters arising from the discretization of 3D Laplacian on a cube decomposed into cube subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m × m cube subdomains connected by face and optionally edge averages increases proportionally to m. The estimates support scalability of unpreconditioned H-FETI-DP (hybrid FETI dual-primal) method. Though the research is most important for the solution of variational inequalities, the results of numerical experiments indicate that unpreconditioned H-FETI-DP with large clusters can be useful also for the solution of huge linear problems.

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