Convergence rates of some iterative methods for nonsymmetric algebraic Riccati equations arising in transport theory

a b s t r a c tThe paper presents a convergence analysis of a modified Newton method for solving nonlinear systems of equations. The convergence results show that this method converges cubically in the nonsingular case, and linearly with the rate 3/8 under some sufficient conditions when the Jacobian is singular at the root. The convergence theory is used to analyze the convergence behavior when the modified Newton method is applied to a nonsymmetric algebraic Riccati equation arising in transport theory. Numerical experiment confirms the theoretical results.

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“…The minimal positive solution of (21) can be obtained via computing the minimal positive solution of the vector equation (22).…”

Section: Application To a Nonsymmetric Algebraic Riccati Equationmentioning

confidence: 99%

“…Here, It has been shown in [22,13] that the Jacobian matrix F (w * ) is a singular M-matrix if and only if the pair of parameters (α, c) = (0, 1).…”

Section: Application To a Nonsymmetric Algebraic Riccati Equationmentioning

confidence: 99%

Convergence analysis of a variant of the Newton method for solving nonlinear equations

Lin

Jia

2010

Computers & Mathematics with Applications

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“…Available iteration algorithms are the Newton's method [4-6, 10, 16, 20], the fixed-point iteration method [1,2,8,14,15] and the structure-preserving doubling algorithm [3,7,9,11,17,19].…”

Section: Introductionmentioning

confidence: 99%

A modified large-scale structure-preserving doubling algorithm for a large-scale Riccati equation from transport theory

Guo

2015

Numer Algor

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A large scale nonsymmetric algebraic Riccati equation XCX−XE−AX+ B = 0 arising in transport theory is considered, where the n × n coefficient matrices B, C are symmetric and low-ranked and A, E are rank one updates of nonsingular diagonal matrices. By introducing a balancing strategy and setting appropriate initial matrices carefully, we can simplify the large-scale structure-preserving doubling algorithm (SDA ls) for this special equation. We give modified large-scale structurepreserving doubling algorithm, which can reduce the flop count of original SDA ls by half. Numerical experiments illustrate the effectiveness of our method.

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“…Symmetric algebraic Riccati equations have been the topic of extensive research, and the theory, applications, and numerical solutions of these equations are the subject of [5,6,7,8] as well as the monographs [21,29]. The minimal positive solution to the NARE (1.1), for medium size problems without the sparseness and low-ranked assumptions, has been studied recently by several authors, employing functional iterations, Newton's method, and the structure-preserving algorithm; see [1,3,4,12,13,14,15,16,17,22,23,26,27,30,31], and the references therein. Evidently, the applications associated with and the numerical solution to NAREs have attracted much attention in the past decade but this paper is the first on general large-scale NAREs.…”

Section: Introduction Consider the Nonsymmetric Algebraic Riccati Eqmentioning

confidence: 99%

Solving Large-Scale Nonsymmetric Algebraic Riccati Equations by Doubling

Li¹,

Chu²,

Kuo³

et al. 2013

SIAM J. Matrix Anal. & Appl.

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We consider the solution of the large-scale nonsymmetric algebraic Riccati equation∈ R (n 1 +n 2 )×(n 1 +n 2 ) being a nonsingular M-matrix. In addition, A and D are sparselike, with the products A −1 u, A − u, D −1 v, and D − v computable in O(n) complexity (with n = max{n 1 , n 2 }), for some vectors u and v, and B, C are low ranked. The structure-preserving doubling algorithms (SDA) by Guo, Lin, and Xu [Numer. Math., 103 (2006), pp. 392-412] is adapted, with the appropriate applications of the Sherman-MorrisonWoodbury formula and the sparse-plus-low-rank representations of various iterates. The resulting large-scale doubling algorithm has an O(n) computational complexity and memory requirement per iteration and converges essentially quadratically. A detailed error analysis, on the effects of truncation of iterates with an explicit forward error bound for the approximate solution from the SDA, and some numerical results will be presented.

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Convergence rates of some iterative methods for nonsymmetric algebraic Riccati equations arising in transport theory

Abstract: We determine and compare the convergence rates of various fixed-point iterations for finding the minimal positive solution of a class of nonsymmetric algebraic Riccati equations arising in transport theory.

Cited by 27 publications

References 15 publications

Convergence analysis of a variant of the Newton method for solving nonlinear equations

Convergence analysis of a variant of the Newton method for solving nonlinear equations

A modified large-scale structure-preserving doubling algorithm for a large-scale Riccati equation from transport theory

Solving Large-Scale Nonsymmetric Algebraic Riccati Equations by Doubling

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