Optimal H2 order-one reduction by solving eigenproblems for polynomial equations

A method is given for solving an optimal H2 approximation problem for SISO linear time-invariant stable systems. The method, based on constructive algebra, guarantees that the global optimum is found; it does not involve any gradient-based search, and hence avoids the usual problems of local minima. We examine mostly the case when the model order is reduced by one, and when the original system has distinct poles. This case exhibits special structure which allows us to provide a complete solution. The problem is converted into linear algebra by exhibiting a finite-dimensional basis for a certain space, and can then be solved by eigenvalue calculations, following the methods developed by Stetter and Moeller. The use of Buchberger's algorithm is avoided by writing the first-order optimality conditions in a special form, from which a Groebner basis is immediately available. Compared with our previous work the method presented here has much smaller time and memory requirements, and can therefore be applied to systems of significantly higher McMillan degree. In addition, some hypotheses which were required in the previous work have been removed. Some examples are included.

On diagonal-Schur complements of block diagonally dominant matrices

It is known that the diagonal-Schur complements of strictly diagonally dominant matrices are strictly diagonally dominant matrices [J.Z. Liu, Y.Q. Huang, Some properties on Schur complements of H-matrices and diagonally dominant matrices, Linear Algebra Appl. 389 (2004) 365-380], and the same is true for nonsingular H-matrices [J.Z. Liu, J.C. Li, Z.T. Huang, X. Kong, Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices, Linear Algebra Appl. 428 (2008) 1009-1030]. In this paper, we research the properties on diagonal-Schur complements of block diagonally dominant matrices and prove that the diagonal-Schur complements of block strictly diagonally dominant matrices are block strictly diagonally dominant matrices, and the same holds for generalized block strictly diagonally dominant matrices. (C) 2010 Elsevier Inc. All rights reserved.

Rank-preserving geometric means of positive semi-definite matrices

The generalization of the geometric mean of positive scalars to positive definite matrices has attracted considerable attention since the seminal work of Ando. The paper generalizes this framework of matrix means by proposing the definition of a rank-preserving mean for two or an arbitrary number of positive semi-definite matrices of fixed rank. The proposed mean is shown to be geometric in that it satisfies all the expected properties of a rank-preserving geometric mean. The work is motivated by operations on low-rank approximations of positive definite matrices in high-dimensional spaces.© 2012 Elsevier Inc. All rights reserved.

Continuous-time subspace flows related to the symmetric eigenproblem

The classes of continuous-time flows on Rn×p that induce the same flow on the set of p- dimensional subspaces of Rn×p are described. The power flow is briefly reviewed in this framework, and a subspace generalization of the Rayleigh quotient flow [Linear Algebra Appl. 368C, 2003, pp. 343-357] is proposed and analyzed. This new flow displays a property akin to deflation in finite time. © 2008 Yokohama Publishers.

on lin-bose problem

This paper study generalized Serre problem proposed by Lin and Bose in multidimensional system theory context [Multidimens. Systems and Signal Process. 10 (1999) 379; Linear Algebra Appl. 338 (2001) 125]. This problem is stated as follows. Let F ∈ Al×m be a full row rank matrix, and d be the greatest common divisor of all the l × l minors of F. Assume that the reduced minors of F generate the unit ideal, where A = K[x 1,...,xn] is the polynomial ring in n variables x 1,...,xn over any coefficient field K. Then there exist matrices G ∈ Al×l and F1 ∈ A l×m such that F = GF1 with det G = d and F 1 is a ZLP matrix. We provide an elementary proof to this problem, and treat non-full rank case.

Two-dimensional Banach spaces with polynomial numerical index zero

We study two-dimensional Banach spaces with polynomial numerical indices equal to zero.

Deployment on GPUs of an application in computational atomic physics

This paper describes the deployment on GPUs of PROP, a program of the 2DRMP suite which models electron collisions with H-like atoms and ions. Because performance on GPUs is better in single precision than in double precision, the numerical stability of the PROP program in single precision has been studied. The numerical quality of PROP results computed in single precision and their impact on the next program of the 2DRMP suite has been analyzed. Successive versions of the PROP program on GPUs have been developed in order to improve its performance. Particular attention has been paid to the optimization of data transfers and of linear algebra operations. Performance obtained on several architectures (including NVIDIA Fermi) are presented.

Efficiently Scheduling Task Dataflow Parallelism: A Comparison Between Swan and QUARK

Increased system variability and irregularity of parallelism in applications put increasing demands on the ef- ficiency of dynamic task schedulers. This paper presents a new design for a work-stealing scheduler supporting both Cilk- style recursively parallel code and parallelism deduced from dataflow dependences. Initial evaluation on a set of linear algebra kernels demonstrates that our scheduler outperforms PLASMA’s QUARK scheduler by up to 12% on a 16-thread Intel Xeon and by up to 50% on a 32-thread AMD Bulldozer.