scholarly journals Convergence of pure and relaxed Newton methods for solving a matrix polynomial equation arising in stochastic models

2014 ◽  
Vol 440 ◽  
pp. 34-49 ◽  
Author(s):  
Jong-Hyeon Seo ◽  
Hyun-Min Kim
Keyword(s):  
Stochastic Models ◽  
Matrix Polynomial ◽  
Polynomial Equation ◽  
Newton Methods ◽  
Matrix Polynomial Equation

On solutions of second-order matrix polynomial equation of high degree

2021 ◽  
pp. 2250100
Author(s):  
Lu Tan ◽  
Xue-Han Cheng ◽  
Tong-Song Jiang ◽  
Si-Tao Ling
Keyword(s):  
Matrix Polynomial ◽  
Sufficient Conditions ◽  
Polynomial Equation ◽  
Second Order ◽  
General Solutions ◽  
Order Matrix ◽  
Algebraic Properties ◽  
Analytic Expressions ◽  
Matrix Polynomial Equation ◽  
High Degree

In this paper, we focus on discussing diagonal solutions and general solutions of second-order matrix polynomial equation of high degree in complex field. By characterizing some algebraic properties of the mentioned two types of the solutions, we present sufficient conditions that a general second-order matrix polynomial equation has diagonal solutions or general solutions. Analytic expressions of the solutions, as well as the corresponding algorithms for finding the solutions are provided. An example is given so as to verify the theoretical results we have derived.

scholarly journals On the Low-Degree Solution of the Sylvester Matrix Polynomial Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Yunbo Tian ◽  
Chao Xia
Keyword(s):  
Matrix Equation ◽  
Matrix Polynomial ◽  
Polynomial Equation ◽  
Sylvester Matrix Equation ◽  
Sylvester Matrix ◽  
Low Degree ◽  
Matrix Polynomial Equation ◽  
Degree Conditions

We study the low-degree solution of the Sylvester matrix equation A 1 λ + A 0 X λ + Y λ B 1 λ + B 0 = C 0 , where A 1 λ + A 0 and B 1 λ + B 0 are regular. Using the substitution of parameter variables λ , we assume that the matrices A 0 and B 0 are invertible. Thus, we prove that if the equation is solvable, then it has a low-degree solution L λ , M λ , satisfying the degree conditions δ L λ < Ind A 0 − 1 A 1  and  δ M λ < Ind B 1 B 0 − 1 .

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