Accurate solutions of M-matrix algebraic Riccati equations

Xue, Jing; Xu, Shengyuan; Li, Ren-Cang

doi:10.1007/s00211-011-0421-0

Cited by 33 publications

(44 citation statements)

References 22 publications

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“…This is the first paper of ours in a sequel of two on accurate solutions of MSE and M -matrix algebraic Riccati equation (1.3) which is more difficult to analyze than MSE because of its nonlinearity in X. This latter will be the subject of our investigation in [26].…”

Section: Discussionmentioning

confidence: 91%

“…See Theorem 3.3. This case will become important later in our perturbation analysis for the Wiener-Hopf factorization in [26]. Proof.…”

Section: Resultsmentioning

confidence: 96%

“…It turns out that if W is entrywise accurate, then the minimal nonnegative solution too is determined and can be computed to a comparable entrywise accuracy. All these will be the subject of study in our forthcoming paper [26], where the perturbation analysis in this article lays the foundation.…”

Section: It Is Called a Nonsingular M -Matrix If γ > ρ(E) And A Singumentioning

confidence: 99%

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Accurate solutions of M-matrix Sylvester equations

Xue

2011

Numer. Math.

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This paper is concerned with a relative perturbation theory and its entrywise relatively accurate numerical solutions of an M -matrix Sylvester equation AX +XB = C by which we mean either both A and B are nonsingular M -matrices or one of them and P = I m ⊗A+B T ⊗I n are nonsingular M -matrices, and C is entrywise nonnegative. It is proved that small relative perturbations to the entries of A, B, and C introduce small relative errors to the entries of the solution X. Thus the smaller entries of X do not suffer bigger relative errors than its larger entries, unlikely the existing perturbation theory for (general) Sylvester equations. We then discuss some minor but crucial implementation changes to three existing numerical methods so that they can be used to compute X as accurately as the input data deserve.

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Section: Discussionmentioning

confidence: 91%

“…See Theorem 3.3. This case will become important later in our perturbation analysis for the Wiener-Hopf factorization in [26]. Proof.…”

Section: Resultsmentioning

confidence: 96%

Section: It Is Called a Nonsingular M -Matrix If γ > ρ(E) And A Singumentioning

confidence: 99%

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Accurate solutions of M-matrix Sylvester equations

Xue

2011

Numer. Math.

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“…Componentwise perturbation analysis [26] shows that small relative perturbations to all entries of A, B, C and D introduces small relative changes to the entries of . Thus smaller entries of do not suffer bigger relative errors than its larger entries.…”

Section: Mathematics Subject Classificationmentioning

confidence: 99%

Highly accurate doubling algorithms for M-matrix algebraic Riccati equations

Xue

2016

Numer. Math.

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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 quadratically convergent, except for MARE in the critical case where convergence is linear with the linear rate 1/2. However, the initialization phase and the doubling iteration kernel of any doubling algorithm involve inverting nonsingular M-matrices. In particular, for MARE in the critical case, the Mmatrices in the doubling iteration kernel, although nonsingular, move towards singular M-matrices at convergence. These inversions are causes of concerns on entrywise relative accuracy of the eventually computed minimal nonnegative solution. Fortunately, a nonsingular M-matrix can be inverted by the GTH-like algorithm of Alfa et al. (Math Comp 71:217-236, 2002) to almost full entrywise relative accuracy, provided a triplet representation of the matrix is known. Recently, Nguyen and Poloni (Numer Math 130(4):763-792, 2015) discovered a way to construct triplet representations in a cancellation-free manner for all involved M-matrices in the doubling iteration kernel, for a special class of MAREs arising from Markov-modulated fluid queues. In this paper, we extend Nguyen's and Poloni's work to all MAREs by also devising a way to construct the triplet representations cancellation-free. Our construction, however, is not a straightforward extension of theirs. It is made possible by an introduction of novel recursively computable auxiliary nonnegative vectors. As the second contribution, we B Ren-Cang Li propose an entrywise relative residual for an approximate solution. The residual has an appealing feature of being able to reveal the entrywise relative accuracies of all entries, large and small, of the approximation. This is in marked contrast to the usual legacy normalized residual which reflects relative accuracies of large entries well but not so much those of very tiny entries. Numerical examples are presented to demonstrate and confirm our claims. Mathematics Subject Classification

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“…In this case, componentwise analysis can be one alternative approach by which much tighter and revealing bounds can be obtained. There are two kinds of alternative condition numbers called mixed and componentwise condition numbers, respectively, which are developed by Gohberg and Koltracht [17], and we refer to [16,22,34,35,[39][40][41][42][43] for more details of these two kinds of condition numbers.…”

Section: Introductionmentioning

confidence: 99%

Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

2018

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We consider a matrix polynomial equation (MPE) A n X n + A n−1 X n−1 + · · · + A 0 = 0, where A n , A n−1 , . . . , A 0 ∈ R m×m are the coefficient matrices, and X ∈ R m×m is the unknown matrix. A sufficient condition for the existence of the minimal nonnegative solution is derived, where minimal means that any other solution is componentwise no less than the minimal one. The explicit expressions of normwise, mixed and componentwise condition numbers of the matrix polynomial equation are obtained. A backward error of the approximated minimal nonnegative solution is defined and evaluated. Some numerical examples are given to show the sharpness of the three kinds of condition numbers.

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Accurate solutions of M-matrix algebraic Riccati equations

Cited by 33 publications

References 22 publications

Accurate solutions of M-matrix Sylvester equations

Accurate solutions of M-matrix Sylvester equations

Highly accurate doubling algorithms for M-matrix algebraic Riccati equations

Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

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