A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential operators and the one-dimensional ultraspherical spectral method. If a partial differential operator is of splitting rank 2, such as the operator associated wi...

In this article, we study methods to solve a Sylvester equation in the form of $\mathbf {A}\mathbf {X}+\mathbf {X}\mathbf {B}=\mathbf {C}$ for given matrices {A}, \mathbf {B}, {C}\in \mathbb {R}^{n\times n}$ ...

This technical report describes three approaches for solving the Differential Riccati Equation (DRE), by means of the Backward Differentiation Formula (BDF) and resolution of the corresponding implicit equation, using Newton's method. These approaches are based on: GMRES method, resolution of Sylvester equation and fixed point method. The role and use of DRE is especially important in optimal c...

Model order reduction (MOR) is common in simulation, control and optimization of complex dynamical systems arising in modeling of physical processes and in the spatial discretization of parabolic partial differential equations (PDEs) in two or more dimensions. Typically, after a semidiscretization of the differential operator by the finite element method (FEM) or by the boundary element method,...

The problem of approximating the greatest common divisor(GCD) for polynomials with inexact coefficients can be formulated as a low rank approximation problem with Sylvester matrix. This paper presents a method based on Structured Total Least Norm(STLN) for constructing the nearest Sylvester matrix of given lower rank. We present algorithms for computing the nearest GCD and a certified 2-GCD for...

The problem of approximating the greatest common divisor(GCD) for multivariate polynomials with inexact coefficients can be formulated as a low rank approximation problem with Sylvester matrix. This paper presents a method based on Structured Total Least Norm(STLN) for constructing the nearest Sylvester matrix of given lower rank. We present algorithms for computing the nearest GCD and a certif...

An alternative simple proof for a well-known result encountered in blind identification and equalization of communication channels is derived. It is shown that the coprimeness condition of a number of FIR channels is equivalent to the full column rank condition of a certain block Sylvester matrix, whose entries are the channel coefficients. As a corollary, it is also shown that in general, the ...