This work is concerned with the numerical solution of large-scale linear matrix equations A1XB T 1 + · · ·+ AKXB K = C. The most straightforward approach computes X ∈ Rm×n from the solution of an mn×mn linear system, typically limiting the feasible values of m,n to a few hundreds at most. Our new approach exploits the fact that X can often be well approximated by a low-rank matrix. It combines ...

In this paper, updating algorithms for solving linear systems of equations are presented using a systolic array model. First, a parallel algorithm for computing the inverse of rank-one modiied matrix using the Sherman-Morrison formula is proposed. This algorithm is then extended to solving the updated systems of linear equations on a linear systolic array. Finally, the generalisation to the upd...

This paper studies systems of polynomial equations that provide information about orientability of matroids. First, we study systems of linear equations over F2, originally alluded to by Bland and Jensen in their seminal paper on weak orientability. The Bland-Jensen linear equations for a matroid M have a solution if and only if M is weakly orientable. We use the Bland-Jensen system to determin...

For a higher-rank graph Λ with sources we detail a construction that creates a row-finite higher-rank graph Λ that does not have sources and contains Λ as a subgraph. Furthermore, when Λ is row-finite the Cuntz-Krieger algebra of Λ, C(Λ) is a full corner of C(Λ), the Cuntz-Krieger algebra of Λ.

Extreme learning machine (ELM) is one of the most popular and important learning algorithms. It comes from single-hidden layer feedforward neural networks. It has been proved that ELM can achieve better performance than support vector machine(SVM) in regression and classification. In this paper, mathematically, with regression problem, the step 3 of ELM is studied. First of all, the equation Hβ...

The randomized Kaczmarz algorithm has received considerable attention recently because of its simplicity, speed, and the ability to approximately solve large-scale linear systems equations. In this paper we propose double triple algorithms extended normal equations form $$\mathbf{A}^\top \mathbf{Ax}=\mathbf{A}^\top \mathbf{b}-\mathbf{c}$$ . proposed avoid forming \mathbf{A}$$ explicitly work fo...

We study the solution of linear systems resulting from the discreitization of unsteady diffusion equations with stochastic coefficients. In particular, we focus on those linear systems that are obtained using the so-called stochastic Galerkin finite element method (SGFEM). These linear systems are usually very large with Kronecker product structure and, thus, solving them can be both timeand co...

with a large matrix A of ill-determined rank. Thus, A has many “tiny” singular values of different orders of magnitude. In particular, A is severely ill-conditioned. Some of the singular values of A may be vanishing. We allow m ≥ n or m < n. The right-hand side vector b̃ is not required to be in the range of A. Linear systems of equations of the form (1) with a matrix of ill-determined rank are ...

We consider multi-echelon supply chains, typically consisting of retailer, distributor, manufacturer, etc. The inventory of each actor is subject to its own delay in delivery time, and each chooses its own inventory replenishment strategy. We model this interaction with a system of linear differential delay equations. When put in matrix form the system is triangular and we derive an exact solut...