Uwe Helmke

Mohsen Zamani, Uwe Helmke, Brian D. O. Anderson

This paper studies zeros of networked linear systems with time-invariant interconnection topology. While the characterization of zeros is given for both heterogeneous and homogeneous networks, homogeneous networks are explored in greater detail. In the current paper, for homogeneous networks with time-invariant interconnection dynamics, it is illustrated how the zeros of each individual agent's system description and zeros definable from the interconnection dynamics contribute to generating zeros of the whole network. We also demonstrate how zeros of networked systems and those of their associated blocked versions are related.

Uwe Helmke, Knut Hüper, Jochen Trumpf

A general class of Newton algorithms on Graßmann and Lagrange-Graßmann manifolds is introduced, that depends on an arbitrary pair of local coordinates. Local quadratic convergence of the algorithm is shown under a suitable condition on the choice of coordinate systems. Our result extends and unifies previous convergence results for Newton's method on a manifold. Using special choices of the coordinates, new numerical algorithms are derived for principal component analysis and invariant subspace computations with improved computational complexity properties.

Brian D. O. Anderson, Shaoshuai Mou, A. Stephen Morse et al.

The paper develops a technique for solving a linear equation $Ax=b$ with a square and nonsingular matrix $A$, using a decentralized gradient algorithm. In the language of control theory, there are $n$ agents, each storing at time $t$ an $n$-vector, call it $x_i(t)$, and a graphical structure associating with each agent a vertex of a fixed, undirected and connected but otherwise arbitrary graph $\mathcal G$ with vertex set and edge set $\mathcal V$ and $\mathcal E$ respectively. We provide differential equation update laws for the $x_i$ with the property that each $x_i$ converges to the solution of the linear equation exponentially fast. The equation for $x_i$ includes additive terms weighting those $x_j$ for which vertices in $\mathcal G$ corresponding to the $i$-th and $j$-th agents are adjacent. The results are extended to the case where $A$ is not square but has full row rank, and bounds are given on the convergence rate.