Computers & Mathematics with Applications
Convergence analysis of a variant of the Newton method for solving nonlinear equations
Computers & Mathematics with Applications
General Matrix Pencil Techniques for Solving Discrete-Time Nonsymmetric Algebraic Riccati Equations
SIAM Journal on Matrix Analysis and Applications
Alternating-directional Doubling Algorithm for $M$-Matrix Algebraic Riccati Equations
SIAM Journal on Matrix Analysis and Applications
On the numerical solution of a structured nonsymmetric algebraic Riccati equation
Performance Evaluation
Monotone convergence of Newton-like methods for M-matrix algebraic Riccati equations
Numerical Algorithms
Hi-index | 0.00 |
We study the nonsymmetric algebraic Riccati equation whose four coefficient matrices are the blocks of a nonsingular $M$-matrix or an irreducible singular $M$-matrix $M$. The solution of practical interest is the minimal nonnegative solution. We show that Newton’s method with zero initial guess can be used to find this solution without any further assumptions. We also present a qualitative perturbation analysis for the minimal solution, which is instructive in designing algorithms for finding more accurate approximations. For the most practically important case, in which $M$ is an irreducible singular $M$-matrix with zero row sums, the minimal solution is either stochastic or substochastic and the Riccati equation can be transformed into a unilateral matrix equation by a procedure of Ramaswami. The minimal solution of the Riccati equation can then be found by computing the minimal nonnegative solution of the unilateral equation using the Latouche-Ramaswami algorithm. When the minimal solution of the Riccati equation is stochastic, we show that the Latouche-Ramaswami algorithm, combined with a shift technique suggested by He, Meini, and Rhee, is breakdown-free and is able to find the minimal solution more efficiently and more accurately than the algorithm without a shift. When the minimal solution of the Riccati equation is substochastic, we show how the substochastic minimal solution can be found by computing the stochastic minimal solution of a related Riccati equation of the same type.