Efficient Approaches for Enclosing the United Solution Set of the Interval Generalized Sylvester Matrix Equation

In this work, we investigate the interval generalized Sylvester matrix equation ${\bf{A}}X{\bf{B}}+{\bf{C}}X{\bf{D}}={\bf{F}}$ and develop some techniques for obtaining outer estimations for the so-called united solution set of this interval system. First, we propose a modified variant of the Krawczyk operator which causes reducing computational complexity to cubic, compared to Kronecker product form. We then propose an iterative technique for enclosing the solution set. These approaches are based on spectral decompositions of the midpoints of ${\bf{A}}$, ${\bf{B}}$, ${\bf{C}}$ and ${\bf{D}}$ and in both of them we suppose that the midpoints of ${\bf{A}}$ and ${\bf{C}}$ are simultaneously diagonalizable as well as for the midpoints of the matrices ${\bf{B}}$ and ${\bf{D}}$. Some numerical experiments are given to illustrate the performance of the proposed methods

Sparsing in Real Time Simulation

Modelling of mechatronical systems often leads to large DAEs with stiff components. In real time simulation neither implicit nor explicit methods can cope with such systems in an efficient way: explicit methods have to employ too small steps and implicit methods have to solve too large systems of equations. A solution of this general problem is to use a method that allows manipulations of the Jacobian by computing only those parts that are necessary for the stability of the method. Specifically, manipulation by sparsing aims at zeroing out certain elements of the Jacobian leading t a structure that can be exploited using sparse matrix techniques. The elements to be neglected are chosen by an a priori analysis phase that can be accomplished before the real-time simulaton starts. In this article a sparsing criterion for the linearly implicit Euler method is derived that is based on block diagnonalization and matrix perturbation theory

Robust isogeometric preconditioners for the Stokes system based on the Fast Diagonalization method

In this paper we propose a new class of preconditioners for the isogeometric discretization of the Stokes system. Their application involves the solution of a Sylvester-like equation, which can be done efficiently thanks to the Fast Diagonalization method. These preconditioners are robust with respect to both the spline degree and mesh size. By incorporating information on the geometry parametrization and equation coefficients, we maintain efficiency on non-trivial computational domains and for variable kinematic viscosity. In our numerical tests we compare to a standard approach, showing that the overall iterative solver based on our preconditioners is significantly faster.Comment: 31 pages, 4 figure

Generalization of Roth's solvability criteria to systems of matrix equations

W.E. Roth (1952) proved that the matrix equation $AX-XB=C$ has a solution if and only if the matrices $\left[\begin{matrix}A&C\\0&B\end{matrix}\right]$ and $\left[\begin{matrix}A&0\\0&B\end{matrix}\right]$ are similar. A. Dmytryshyn and B. K{\aa}gstr\"om (2015) extended Roth's criterion to systems of matrix equations $A_iX_{i'}M_i-N_iX_{i''}^{\sigma_i} B_i=C_i$ $(i=1,\dots,s)$ with unknown matrices $X_1,\dots,X_t$, in which every $X^{\sigma}$ is $X$, $X^T$, or $X^*$. We extend their criterion to systems of complex matrix equations that include the complex conjugation of unknown matrices. We also prove an analogous criterion for systems of quaternion matrix equations.Comment: 11 page

Block-diagonalisation of matrices and operators

In this short note we deal with a constructive scheme to decompose a continuous family of matrices $A(\rho)$ asymptotically as $\rho\to0$ into blocks corresponding to groups of eigenvalues of the limit matrix A(0). We also discuss the extension of the scheme to matrix families depending upon additional parameters and operators on Hilbert spaces.Comment: 8 page

Solvability and uniqueness criteria for generalized Sylvester-type equations

We provide necessary and sufficient conditions for the generalized $\star$-Sylvester matrix equation, $AXB + CX^\star D = E$, to have exactly one solution for any right-hand side E. These conditions are given for arbitrary coefficient matrices $A, B, C, D$ (either square or rectangular) and generalize existing results for the same equation with square coefficients. We also review the known results regarding the existence and uniqueness of solution for generalized Sylvester and $\star$-Sylvester equations.Comment: This new version corrects some inaccuracies in corollaries 7 and

A nested divide-and-conquer method for tensor Sylvester equations with positive definite hierarchically semiseparable coefficients

Linear systems with a tensor product structure arise naturally when considering the discretization of Laplace type differential equations or, more generally, multidimensional operators with separable coefficients. In this work, we focus on the numerical solution of linear systems of the form $$\left(I\otimes \dots\otimes I \otimes A_1+\dots + A_d\otimes I \otimes\dots \otimes I\right)x=b,$$ where the matrices $A_t\in\mathbb R^{n\times n}$ are symmetric positive definite and belong to the class of hierarchically semiseparable matrices. We propose and analyze a nested divide-and-conquer scheme, based on the technology of low-rank updates, that attains the quasi-optimal computational cost $\mathcal O(n^d (\log(n) + \log(\kappa)^2 + \log(\kappa) \log(\epsilon^{-1})))$ where $\kappa$ is the condition number of the linear system, and $\epsilon$ the target accuracy. Our theoretical analysis highlights the role of inexactness in the nested calls of our algorithm and provides worst case estimates for the amplification of the residual norm. The performances are validated on 2D and 3D case studies

Exact solution of the Schrodinger equation with the spin-boson Hamiltonian

We address the problem of obtaining the exact reduced dynamics of the spin-half (qubit) immersed within the bosonic bath (enviroment). An exact solution of the Schrodinger equation with the paradigmatic spin-boson Hamiltonian is obtained. We believe that this result is a major step ahead and may ultimately contribute to the complete resolution of the problem in question. We also construct the constant of motion for the spin-boson system. In contrast to the standard techniques available within the framework of the open quantum systems theory, our analysis is based on the theory of block operator matrices.Comment: 9 pages, LaTeX, to appear in Journal of Physics A: Mathematical and Theoretica

Roth’s solvability criteria for the matrix equations AX - XB^ = C and X - AXB^ = C over the skew field of quaternions with aninvolutive automorphism q ¿ qˆ

The matrix equation AX-XB = C has a solution if and only if the matrices A C 0 B and A 0 0 B are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) proved that the matrix equation X - AXB = C over a field has a solution if and only if the matrices A C 0 I and I 0 0 B are simultaneously equivalent to A 0 0 I and I 0 0 B . We extend these criteria to the matrix equations AX- ^ XB = C and X - A ^ XB = C over the skew field of quaternions with a fixed involutive automorphism q ¿ ˆq.Postprint (author's final draft

Autonomous Volterra algorithm for steady-state analysis of nonlinear circuits

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