Nonnegative Solutions of a Quadratic Matrix Equation Arising from Comparison Theorems in Ordinary Differential Equations

1985 ◽  
Vol 6 (1) ◽  
pp. 47-53 ◽  
Author(s):  
G. J. Butler ◽  
Charles R. Johnson ◽  
H. Wolkowicz
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Matrix Equation ◽  
Comparison Theorems ◽  
Nonnegative Solutions ◽  
Quadratic Matrix Equation ◽  
Quadratic Matrix

Newton-Shamanskii Method for a Quadratic Matrix Equation Arising in Quasi-Birth-Death Problems

2014 ◽  
Vol 4 (4) ◽  
pp. 386-395
Author(s):  
Pei-Chang Guo
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Matrix Equation ◽  
Nonnegative Solution ◽  
Minimal Nonnegative Solution ◽  
The Minimal Nonnegative Solution ◽  
Quadratic Matrix Equation ◽  
Quadratic Matrix ◽  
Birth Death

AbstractIn order to determine the stationary distribution for discrete time quasi-birth-death Markov chains, it is necessary to find the minimal nonnegative solution of a quadratic matrix equation. The Newton-Shamanskii method is applied to solve this equation, and the sequence of matrices produced is monotonically increasing and converges to its minimal nonnegative solution. Numerical results illustrate the effectiveness of this procedure.

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