A Matrix-Form LSQR Iterative Method for Solving the Second-Order Sylvester Matrix Equation EXF2+AXF+CX+BY=D

2010 ◽  
Author(s):  
Sheng-Kun Li
Keyword(s):  
Iterative Method ◽  
Matrix Equation ◽  
Second Order ◽  
Matrix Form ◽  
Sylvester Matrix Equation ◽  
Sylvester Matrix

scholarly journals Modified Jacobi-Gradient Iterative Method for Generalized Sylvester Matrix Equation

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1831
Author(s):  
Nopparut Sasaki ◽  
Pattrawut Chansangiam
Keyword(s):  
Iterative Method ◽  
Matrix Equation ◽  
Recursive Formula ◽  
Sylvester Matrix Equation ◽  
Hierarchical Identification ◽  
Sylvester Matrix ◽  
Generalized Sylvester Matrix Equation ◽  
Iteration Matrix ◽  
Diagonal Part ◽  
Gradient Based

We propose a new iterative method for solving a generalized Sylvester matrix equation A1XA2+A3XA4=E with given square matrices A1,A2,A3,A4 and an unknown rectangular matrix X. The method aims to construct a sequence of approximated solutions converging to the exact solution, no matter the initial value is. We decompose the coefficient matrices to be the sum of its diagonal part and others. The recursive formula for the iteration is derived from the gradients of quadratic norm-error functions, together with the hierarchical identification principle. We find equivalent conditions on a convergent factor, relied on eigenvalues of the associated iteration matrix, so that the method is applicable as desired. The convergence rate and error estimation of the method are governed by the spectral norm of the related iteration matrix. Furthermore, we illustrate numerical examples of the proposed method to show its capability and efficacy, compared to recent gradient-based iterative methods.

scholarly journals On the Parametric Solution to the Second-Order Sylvester Matrix EquationEVF2−AVF−CV=BW

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
Wang Guo-Sheng ◽  
Lv Qiang ◽  
Duan Guang-Ren
Keyword(s):  
Control Systems ◽  
Matrix Equation ◽  
Second Order ◽  
Diagonal Form ◽  
Sylvester Matrix Equation ◽  
Parametric Solution ◽  
Design Problems ◽  
Analysis And Design ◽  
Sylvester Matrix ◽  
The Matrix

This paper considers the solution to a class of the second-order Sylvester matrix equationEVF2−AVF−CV=BW. Under the controllability of the matrix triple(E,A,B), a complete, general, and explicit parametric solution to the second-order Sylvester matrix equation, with the matrixFin a diagonal form, is proposed. The results provide great convenience to the analysis of the solution to the second-order Sylvester matrix equation, and can perform important functions in many analysis and design problems in control systems theory. As a demonstration, an illustrative example is given to show the effectiveness of the proposed solution.

scholarly journals Classification and approximation of solutions to Sylvester matrix equation

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4261-4280 ◽  
Author(s):  
Bogdan Djordjevic ◽  
Nebojsa Dincic
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Infinitely Many Solutions ◽  
Sylvester Matrix Equation ◽  
Minimal Norm ◽  
Sylvester Matrix

In this paperwesolve Sylvester matrix equation with infinitely-many solutions and conduct their classification. If the conditions for their existence are not met, we provide a way for their approximation by least-squares minimal-norm method.

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