Both Sylvester matrix and convolution are defined by two polynomials. If one of them has small degree, then the associated Sylvester linear system can be solved fast by using its PFD interpretation of the convolution equation. This can immediately simplify the refinement of approximate convolution by means of Newton’s iteration, where we also incorporate the PFD refinement techniques or alterna...

Global Krylov subspace methods are the most efficient and robust methods to solve generalized coupled Sylvester matrix equation. In this paper, we propose the nested splitting conjugate gradient process for solving this equation. This method has inner and outer iterations, which employs the generalized conjugate gradient method as an inner iteration to approximate each outer iterate, while each...

Presents a recurrent neural network for solving the Sylvester equation with time-varying coefficient matrices. The recurrent neural network with implicit dynamics is deliberately developed in the way that its trajectory is guaranteed to converge exponentially to the time-varying solution of a given Sylvester equation. Theoretical results of convergence and sensitivity analysis are presented to ...

Both Sylvester matrix and convolution are defined by two polynomials. If one of them has small degree, then the associated Sylvester linear system can be solved fast by using its PFD interpretation of the convolution equation. This can immediately simplify the refinement of approximate convolution by means of Newton’s iteration, where we also incorporate the PFD refinement techniques or alterna...

We consider an analytic perturbation of the Sylvester matrix equation. Mainly we are interested in the singular case, that is, when the null space of the unperturbed Sylvester operator is not trivial, but the perturbed equation has a unique solution. In this case, the solution of the perturbed equation can be given in terms of a Laurent series. Here we provide a necessary and su cient condition...

Descriptor systems consisting of a large number of differential-algebraic equations (DAEs) usually arise from the discretization of partial differential-algebraic equations. This paper presents an efficient algorithm for solving the coupled Sylvester equation that arises in converting a system of linear DAEs to ordinary differential equations. A significant computational advantage is obtained b...

We consider the solution of the ?-Sylvester equation AX±X?B? = C, for ? = T,H and A,B,∈ Cm×n, and some related linear matrix equations (AXB? ± X? = C, AXB? ± CX?D? = E, AX ± X?A? = C, AX ± Y B = C, AXB ± CY D = E, AXA? ± BY B? = C and AXB ± (AXB)? = C). Solvability conditions and stable numerical methods are considered, in terms of the (generalized and periodic) Schur, QR and (generalized) sing...

We provide a formula for variational quasi-Newton updates with multiple weighted secant equations. The derivation of the formula leads to a Sylvester equation in the correction matrix. Examples are given.

We try to arm Newton’s iteration for univariate polynomial factorization with greater convergence power by shifting to a larger basic system of multivariate constraints. The convolution equation is a natural means for a desired expansion of the basis for this iteration versus the classical univariate method, which is more vulnerable to foreign distractions from its convergence course. Compared ...

We try to arm Newton’s iteration for univariate polynomial factorization with greater convergence power by shifting to a larger basic system of multivariate constraints. The convolution equation is a natural means for a desired expansion of the basis for this iteration versus the classical univariate method, which is more vulnerable to foreign distractions from its convergence course. Compared ...