Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNonsingular systems of generalized Sylvester equations: an algorithmic approach
105
0
0.0
(
0
)
Added by
Leonardo Robol
Publication date
2017
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We consider the uniqueness of solution (i.e., nonsingularity) of systems of $r$ generalized Sylvester and $star$-Sylvester equations with $ntimes n$ coefficients. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized $star$-Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix pencils or formal matrix products, both constructed from the coefficients of the system. The proposed approach uses the periodic Schur decomposition, and leads to a backward stable $O(n^3r)$ algorithm for computing the (unique) solution.
rate research
Read More
We present a circulant and skew-circulant splitting (CSCS) iterative method for solving large sparse continuous Sylvester equations $AX + XB = C$, where the coefficient matrices $A$ and $B$ are Toeplitz matrices. A theoretical study shows that if the circulant and skew-circulant splitting factors of $A$ and $B$ are positive semi-definite and at least one is positive definite (not necessarily Hermitian), then the CSCS method converges to the unique solution of the Sylvester equation. In addition, we obtain an upper bound for the convergence factor of the CSCS iteration. This convergence factor depends only on the eigenvalues of the circulant and skew-circulant splitting matrices. A computational comparison with alternative methods reveals the efficiency and reliability of the proposed method.
This paper introduces and analyzes a preconditioned modified of the Hermitian and skew-Hermitian splitting (PMHSS). The large sparse continuous Sylvester equations are solved by PMHSS iterative algorithm based on nonHermitian, complex, positive definite/semidefinite, and symmetric matrices. We prove that the PMHSS is converged under suitable conditions. In addition, we propose an accelerated PMHSS method consisting of two preconditioned matrices and two iteration parameters {alpha}, b{eta}. Theoretical analysis showed that the convergence speed of the accelerated PMHSS is faster compared to the PMHSS. Also, the robustness and efficiency of the proposed two iterative algorithms were demonstrated in numerical experiments.
Techniques from numerical bifurcation theory are very useful to study transitions between steady fluid flow patterns and the instabilities involved. Here, we provide computational methodology to use parameter continuation in determining probability density functions of systems of stochastic partial differential equations near fixed points, under a small noise approximation. Key innovation is the efficient solution of a generalized Lyapunov equation using an iterative method involving low-rank approximations. We apply and illustrate the capabilities of the method using a problem in physical oceanography, i.e. the occurrence of multiple steady states of the Atlantic Ocean circulation.
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.
We consider linear systems $Ax = b$ where $A in mathbb{R}^{m times n}$ consists of normalized rows, $|a_i|_{ell^2} = 1$, and where up to $beta m$ entries of $b$ have been corrupted (possibly by arbitrarily large numbers). Haddock, Needell, Rebrova and Swartworth propose a quantile-based Random Kaczmarz method and show that for certain random matrices $A$ it converges with high likelihood to the true solution. We prove a deterministic version by constructing, for any matrix $A$, a number $beta_A$ such that there is convergence for all perturbations with $beta < beta_A$. Assuming a random matrix heuristic, this proves convergence for tall Gaussian matrices with up to $sim 0.5%$ corruption (a number that can likely be improved).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
