Convergence Analysis of the Doubling Algorithm for Several Nonlinear Matrix Equations in the Critical Case

@article{Chiang2009ConvergenceAO,
  title={Convergence Analysis of the Doubling Algorithm for Several Nonlinear Matrix Equations in the Critical Case},
  author={Chun-Yueh Chiang and Eric King-wah Chu and Chun-Hua Guo and Tsung-Ming Huang and Wen-Wei Lin and Shufang Xu},
  journal={SIAM J. Matrix Anal. Appl.},
  year={2009},
  volume={31},
  pages={227-247}
}
In this paper, we review two types of doubling algorithm and some techniques for analyzing them. We then use the techniques to study the doubling algorithm for three different nonlinear matrix equations in the critical case. We show that the convergence of the doubling algorithm is at least linear with rate $1/2$. As compared to earlier work on this topic, the results we present here are more general, and the analysis here is much simpler. 
View via Publisher
ourspace.uregina.ca
96 Citations
Highly Influential Citations
3
Background Citations
33
Methods Citations
19
Results Citations
8

96 Citations

A structured doubling algorithm for Lur'e Equations

We introduce a numerical method for the numerical solution of the Lur’e equations, a system of matrix equations that arises, for instance, in linear-quadratic infinite time horizon optimal control.…

THE SHIFTING TECHNIQUE FOR SOLVING A NONSYMMETRIC ALGEBRAIC RICCATI EQUATION

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.

The convergence analysis of an accelerated iteration for solving algebraic Riccati equations

A Structured Doubling Algorithm for the Numerical Solution of Lur’e Equations

We introduce a~numerical method for the numerical solution of the Lur'e matrix equations that arise, for instance, in linear-quadratic infinite time horizon optimal control. The method is based on…

Numerical Study on Nonsymmetric Algebraic Riccati Equations

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

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 Cyclic Reduction Algorithm for A Quadratic Matrix Equation Arising from Overdamped Systems

A fast cyclic reduction algorithm is proposed to calculate the extreme solutions of the overdamped quadratic matrix equations arising from the overdamped mass-spring system and it is shown that the proposed algorithm outperforms the originalcyclic reduction and the structure-preserving doubling algorithm.

Balanced truncation-rational Krylov methods for model reduction in large scale dynamical systems

In this paper, we consider the balanced truncation method for model reductions in large-scale linear and time-independent dynamical systems with multi-inputs and multi-outputs. The method is based on…

A generalized structured doubling algorithm for the numerical solution of linear quadratic optimal control problems

A generalization of the structured doubling algorithm to compute invariant subspaces of structured matrix pencils that arise in the context of solving linear quadratic optimal control problems to attain better accuracy.

An accelerated technique for solving the positive definite solutions of a class of nonlinear matrix equations

...

References

SHOWING 1-10 OF 48 REFERENCES

Iterative solution of two matrix equations

Newton's method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation andumerical results are reported to illustrate the convergence behaviour of various algorithms.

Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations

A unified convergence theory for the structure-preserving doubling algorithms for a class of Riccati-type matrix equations is established, using only elementary matrix theory.

A new class of nonsymmetric algebraic Riccati equations

On the Iterative Solution of a Class of Nonsymmetric Algebraic Riccati Equations

Newton's method and a class of basic fixed-point iterations can be used to find its minimal positive solution whenever it has a positive solution of any equation in this class of nonsymmetric algebraic Riccati equations.

Structured doubling algorithms for weakly stabilizing Hermitian solutions of algebraic Riccati equations

Structure-Preserving Algorithms for Periodic Discrete-Time Algebraic Riccati Equations

The structure-preserving doubling algorithm is developed from a new point of view and shows its quadratic convergence under assumptions which are weaker than stabilizability and detectability, and is shown to be efficient, out-performing other algorithms on a large set of benchmark problems.

On the existence of a positive definite solution of the matrix equation

An efficient and numerically stable algorithm for computing the positive definite solution of the nonlinear equation and some properties of the solution are discussed as well as the sufficient conditions for the existence are obtained.

Existence of algebraic matrix Riccati equations arising in transport theory

On the solution of algebraic Riccati equations arising in fluid queues

Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X + A*X-1A = Q