Abstra t. The need to ompute small on-eigenvalues and the asso iated on-eigenve tors of positive-de nite Cau hy matri es naturally arises when onstru ting rational approximations with a (near) optimally small L error. Spe i ally, given a rational fun tion with n poles in the unit disk, a rational approximation with m ≪ n poles in the unit disk may be obtained from the mth on-eigenve tor of an n...

It is a classical inequality that the minimum of the ratio of the (weighted) arithmetic mean to the geometric mean of a set of positive variables is equal to one, and is attained at the center of the positivity cone. While there are numerous proofs of this fundamental homogeneous inequality, in the presence of an arbitrary subspace, and/or the replacement of the arithmetic mean with an arbitrar...

It is shown that an innnite Hankel matrix of a nite rank (or a nite Hankel matrix) admits a generalized Vandermonde decomposition H = V T DV , where V is a generalized Vandermonde matrix, and D is a block diagonal matrix. The full structure of this decomposition was rst fully discussed by Vandevoorde 9], but the development here is based solely on linear algebra considerations, speciically the ...

Many reinforcement learning algorithms, like Q-Learning or R-Learning, correspond to adaptative methods for solving Markovian decision problems in innnite-horizon when no model is available. In this article we consider the particular framework of non-stationary nite-horizon Markov Decision Processes. After establishing a relationship between the nite-horizon total reward criterion and the avera...

A number of mathematicians have considered the problem of writing an operator as a product of \nice" operators, such as positive, hermitian or normal operators. Our principal reference for this is a paper of P.Y. Wu 6], but see also 2] and 5]. This kind of question, and related questions, have also been considered in a C*-algebra context, see 3]. A core result of Wu's paper is his theorem that ...

{ In subspace methods for system identiication, the system matrices can be estimated from least squares, based on estimated Kalman lter state sequences and the observed inputs and outputs. It is well known that for an innnite amount of data, this least squares estimate of the system matrices is unbiased, when the system order is correctly estimated. However, for a nite amount of data, the estim...

Consider the solution of a large linear system when the coe cient matrix is sparse, symmetric, and positive de nite. One approach is the method of \conjugate gradient" (CG) with an incomplete Cholesky (IC) preconditioner (ICCG). A key problem with the design of a parallel ICCG solver is the bottleneck posed by repeated parallel sparse triangular solves to apply the preconditioner. Our work conc...

where the matrix a(y) = [ak`(y)] is symmetric and positive de­ nite a(y) > ¬ I with ¬ > 0. Its entries belong to L # (Y ) (Y is the cube ]0; 2 o [ N and subscript `#’ means that the space consists of Y -periodic functions). In the sequel, we will make various further length-scale regularity assumptions on the coe ̄ cients ak` which play a central role throughout the paper. Some of them are as fo...