Discrete-time symmetric polynomial equations with complex coefficients are studied in the scalar and matrix case. New theoretical results are derived and several algorithms are proposed and evaluated. Polynomial reduction algorithms are first described to study theoretical properties of the equations. Sylvester matrix algorithms are then developed to solve numerically the equations. The algorit...

Acknowledgements: During the research that led to the this paper, the first author was initially supported by the European Regional Development Fund through the programme  and by the PortugueseGovernment through the  (Fundação para a Ciência e a Tecnologia) under the project -/// and through an  Ciência  fellowship, and later supported by an  Investigador a...

The delay Lyapunov equation is an important matrix boundary-value problem which arises as an analogue of the Lyapunov equation in the study of time-delay systems ẋ(t) = A0x(t) +A1x(t− τ) +B0u(t). We propose a new algorithm for the solution of the delay Lyapunov equation. Our method is based on the fact that the delay Lyapunov equation can be expressed as a linear system of equations, whose unkn...

in recent years, there has been greater attempt to find numerical solutions of differential equations using wavelet's methods. the following method is based on vector forms of haar-wavelet functions. in this paper, we will introduce one dimensional haar-wavelet functions and the haar-wavelet operational matrices of the fractional order integration. also the haar-wavelet operational matrice...

We consider here a two sided interpolation problem where we want to minimize the degree of the interpolant. We show that this degree is given by the rank of a particular solution to a Sylvester equation which, in some particular cases becomes a Löwner or a Hankel matrix. We consider an application to the usual partial realization problem. The results are quite general and no particular assumpti...

We consider the uniqueness of solution (nonsingularity) of systems of r generalized Sylvester and ⋆-Sylvester equations with n × n coefficient matrices. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized ⋆-Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix penc...

We consider the numerical approximation to the solution of the matrix equation A1X+XA2 −Y C = 0 in the unknown matrices X, Y , under the constraint XB = 0, with A1, A2 of large dimensions. We propose a new formulation of the problem that entails the numerical solution of an unconstrained Sylvester equation. The spectral properties of the resulting coefficient matrices call for appropriately des...

We introduce Gaussian Process Topic Models (GPTMs), a new family of topic models which can leverage a kernel among documents while extracting correlated topics. GPTMs can be considered a systematic generalization of the Correlated Topic Models (CTMs) using ideas from Gaussian Process (GP) based embedding. Since GPTMs work with both a topic covariance matrix and a document kernel matrix, learnin...

The main is to develop a method to solve an arbitrary fuzzy matrix equation system by using the embedding approach. Considering the existing solution to n × n fuzzy matrix equation system is done. To illustrate the proposed model a numerical example is given, and obtained results are discussed.