On hermitian solutions of the symmetric algebraic Riccati equation
SIAM Journal on Control and Optimization
Analysis and modification of Newton's method for algebraic Riccati equations
Mathematics of Computation
On a Newton-Like Method for Solving Algebraic Riccati Equations
SIAM Journal on Matrix Analysis and Applications
A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations
SIAM Journal on Scientific Computing
Low Rank Solution of Lyapunov Equations
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations
Applied Numerical Mathematics
Fast Iterative Schemes for Nonsymmetric Algebraic Riccati Equations Arising from Transport Theory
SIAM Journal on Scientific Computing
Inexact Kleinman-Newton Method for Riccati Equations
SIAM Journal on Matrix Analysis and Applications
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Numerical methods for solving the symmetric algebraic Riccati equation by using Newton's method are considered in this paper. Instead of direct methods, a fast doubling iteration scheme is applied to inexactly solve the Lyapunov equations arising in each Newton iteration. Then, a new inexact Newton method is proposed by using the Newton iteration as the outer iteration and the doubling iteration as the inner iteration. By controlling the inner iteration for each Newton iteration step, we prove the monotonicity and global convergence of the inexact Newton method. The efficiency of these methods are illustrated by several numerical examples.