This paper presents some inequalities on generalized Schur complements. Let A be an n n (Hermitian) positive semide®nite matrix. Denote by A=a the generalized Schur complement of a principal submatrix indexed by a set a in A. Let A‡ be the Moore± Penrose inverse of A and k…A† be the eigenvalue vector of A. The main results of this paper are: 1. k…A‡…a0††P k……A=a†‡†, where a0 is the complement o...

Maximum likelihood estimates of parameters of some spatial models require the computation of the log-determinant of positive-de®nite matrices of the form Iÿ aD, where D is a large, sparse matrix with eigenvalues in ‰ÿ1; 1Š and where 0 < a < 1. With extremely large matrices the usual direct methods of obtaining the log-determinant require too much time and memory. We propose a Monte Carlo estima...

ABSTRACT Although the stability of ir uit analysis based on partial relu tan e K has been proved for parallel bus stru tures [5℄ and the stru tures with suÆ iently dis retized ondu tors [4℄, the stability for general inter onne t topologies has not been proved. This paper proved that full partial relu tan e matrix and sparsi ed partial relu tan e matrix obtained by ignoring small mutual terms i...

In 2008 all ECB publications feature a motif taken from the €10 banknote. 1 We would like to thank Lucrezia Reichlin for her comments on an earlier draft. Any errors are the authors own. Fax +49 69 1344 6000 All rights reserved. Any reproduction, publication and reprint in the form of a different publication, whether printed or produced electronically, in whole or in part, is permitted only wit...

We consider the iterative solution of symmetric positive-de nite linear systems whose coe cient matrix may be expressed as the outer-product of low-rank terms. We derive suitable preconditioners for such systems, and demonstrate their e ectiveness on a number of test examples. We also consider combining these methods with existing techniques to cope with the commonly-occuring case where the coe...

The conjugate gradient algorithm (CG) is an eeective tool for solving a system of linear equation with a positive deenite coeecient matrix. We show the reasons for a possible breakdown of the method when applied to a symmetric system with an indeenite coeecient matrix. Although in nite arithmetic a breakdown of the method occurs rather seldom, near breakdowns may slow down the speed of converge...

Sparse linear systems Kx = b are considered where K is a specially structured symmetric indeenite matrix. These systems arise frequently, e.g., from mixed nite element discretiza-tions of PDE problems. The LDL T factorization of K with diagonal D and unit lower triangular L is known to exist for natural ordering of K but the resulting triangular factors can be rather dense. On the other hand, f...

We introduce a generalized version of a q-Schur algebra (of parabolic type) for arbitrary Hecke algebras over extended Weyl groups. We describe how the decomposition matrix of a nite group with split BN-pair, with respect to a non-describing prime, can be partially described by the decomposition matrices of suitably chosen q-Schur algebras. We show that the investigated structures occur natural...

We consider a two-level system, which couples via non-commuting operators to two independent oscillator baths. When the coupling is symmetric, the renormalized hopping matrix element is nite even for in nitely strong coupling strength. The two-level system is in a delocalized phase. For nite coupling strength a localization transition occurs for a critical asymmetry angle, which separates the l...

The eigenvectors of a symmetric matrix can be chosen to form a biorthogonal set with respect to the identity and to the matrix itself. Similarly, the eigenvectors of a symmetric de nite linear pencil can be chosen to be biorthogonal with respect to the pair.This paper presents the three sets of matrix weights, with respect to which the eigenvectors of the symmetric de nite quadratic pencil are ...