# Iterative Solution of a Nonsymmetric Algebraic Riccati Equation

@article{Guo2007IterativeSO, title={Iterative Solution of a Nonsymmetric Algebraic Riccati Equation}, author={Chun-Hua Guo and Nicholas John Higham}, journal={SIAM J. Matrix Anal. Appl.}, year={2007}, volume={29}, pages={396-412} }

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…

## 94 Citations

### Numerical Study on Nonsymmetric Algebraic Riccati Equations

- Mathematics
- 2016

In this paper, we consider the nonsymmetric algebraic Riccati equation whose four coefficient matrices form an M-matrix K. When K is a regular M-matrix, the Riccati equation is known to have a…

### Numerical Study on Nonsymmetric Algebraic Riccati Equations

- MathematicsMediterranean Journal of Mathematics
- 2016

In this paper, we consider the nonsymmetric algebraic Riccati equation whose four coefficient matrices form an M-matrix K. When K is a regular M-matrix, the Riccati equation is known to have a…

### A Fast Newton's Method for a Nonsymmetric Algebraic Riccati Equation

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2008

The original equation is transformed into an equivalent Riccati equation where the singularity is removed while the matrix coefficients maintain the same structure as in the original equation, leading to a quadratically convergent algorithm with complexity $O(n^2)$ which provides approximations with full precision.

### Highly accurate doubling algorithms for M-matrix algebraic Riccati equations

- MathematicsNumerische Mathematik
- 2016

The doubling algorithms are very efficient iterative methods for computing the unique minimal nonnegative solution to an M-matrix algebraic Riccati equation (MARE). They are globally and…

### A subspace shift technique for nonsymmetric algebraic Riccati equations associated with an M‐matrix

- MathematicsNumer. Linear Algebra Appl.
- 2013

This paper introduces a new method to accelerate the convergence properties of the iterations also in close-to-critical cases, by working on the invariant subspace associated with the problematic eigenvalues as a whole, and presents a theoretical analysis and several numerical experiments which confirm the efficiency of the new method.

### Sensitivity of the Solution to Nonsymmetric Differential Matrix Riccati Equation

- MathematicsMathematics
- 2021

Nonsymmetric differential matrix Riccati equations arise in many problems related to science and engineering. This work is focusing on the sensitivity of the solution to perturbations in the matrix…

### New Alternately Linearized Implicit Iteration for M-matrix Algebraic Riccati Equations

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- 2017

Research on the theories and the efficient numerical methods of M-matrix algebraic Riccati equation (MARE) has become a hot topic in recent years due to its broad applications. In this paper, based…

### A class of complex nonsymmetric algebraic Riccati equations associated with H-matrix

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### THE SHIFTING TECHNIQUE FOR SOLVING A NONSYMMETRIC ALGEBRAIC RICCATI EQUATION

- Computer Science, Mathematics
- 2011

The main trust in this work is to show that applied with a suitable shifted technique, the SDA is guaranteed to converge quadratically with no breakdown, and modify the conventional simple iteration algorithm in the critical case to dramatically improve the speed of convergence.

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