Fast verified computation for the minimal nonnegative solution of the nonsymmetric algebraic Riccati equation

2018 ◽  
Vol 37 (4) ◽  
pp. 4599-4610 ◽  
Author(s):  
Shinya Miyajima
Keyword(s):  
Riccati Equation ◽  
Nonnegative Solution ◽  
Algebraic Riccati Equation ◽  
Minimal Nonnegative Solution ◽  
The Minimal Nonnegative Solution ◽  
Nonsymmetric Algebraic Riccati Equation ◽  
Verified Computation

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.

Convergence Rates of a Class of Predictor-Corrector Iterations for the Nonsymmetric Algebraic Riccati Equation Arising in Transport Theory

2017 ◽  
Vol 9 (4) ◽  
pp. 944-963 ◽  
Author(s):  
Ning Dong ◽  
Jicheng Jin ◽  
Bo Yu
Keyword(s):  
Riccati Equation ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Transport Theory ◽  
Algebraic Riccati Equation ◽  
Asymptotic Convergence ◽  
Comparison Theory ◽  
Nonsymmetric Algebraic Riccati Equation ◽  
Predictor Corrector ◽  
Asymptotic Convergence Rate

AbstractIn this paper, we analyse the convergence rates of several different predictor-corrector iterations for computing the minimal positive solution of the nonsymmetric algebraic Riccati equation arising in transport theory. We have shown theoretically that the new predictor-corrector iteration given in [Numer. Linear Algebra Appl., 21 (2014), pp. 761–780] will converge no faster than the simple predictor-corrector iteration and the nonlinear block Jacobi predictor-corrector iteration. Moreover the last two have the same asymptotic convergence rate with the nonlinear block Gauss-Seidel iteration given in [SIAM J. Sci. Comput., 30 (2008), pp. 804–818]. Preliminary numerical experiments have been reported for the validation of the developed comparison theory.

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