Transforming algebraic Riccati equations into unilateral quadratic matrix equations

Бини, Дарио Андреа; Meini, Beatrice; Poloni, Federico

doi:10.1007/s00211-010-0319-2

Cited by 46 publications

(70 citation statements)

References 26 publications

(69 reference statements)

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“…The relation between the Riccati and the quadratic matrix equation is highlighted in [7], whereas a study on the existence of solvents can be found in [13]. Several works address the problem of computing a numerical approximation for the solution of the quadratic matrix equation: an approach to compute, when possible, the dominant solvent is proposed in [12].…”

Section: P (S) := J=0mentioning

confidence: 99%

A contour integral approach to the computation of invariant pairs

Barkatou

Boito

Ugalde

2017

Theoretical Computer Science

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We study some aspects of the invariant pair problem for matrix polynomials, as introduced by Betcke and Kressner [3] and by Beyn and Thümmler [6]. Invariant pairs extend the notion of eigenvalue-eigenvector pairs, providing a counterpart of invariant subspaces for the nonlinear case. We compute formulations for the condition numbers and the backward error for invariant pairs and solvents. We then adapt the Sakurai-Sugiura moment method [1] to the computation of invariant pairs, including some classes of problems that have multiple eigenvalues. Numerical refinement via a variant of Newton's method is also studied. Furthermore, we investigate the relation between the matrix solvent problem and the triangularization of matrix polynomials.

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Section: P (S) := J=0mentioning

confidence: 99%

A contour integral approach to the computation of invariant pairs

Barkatou

Boito

Ugalde

2017

Theoretical Computer Science

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“…Indeed, there are several algorithms that, if implemented correctly, can compute such to the accuracy as the theory predicts. They include several fix-point iterations [26,27] and the doubling algorithms [6,15,24]. The fixpoint iterations [26] do not present much of implementation challenges as oppose to the doubling algorithms whose implementations involve inverting many nonsingular M-matrices, which, if not done carefully, leads to loss of accuracy in the tiny entries of the minimal nonnegative solution.…”

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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“…The problem of reducing an algebraic Riccati equation to (1.2) has been analyzed in [4]. Numerical methods for solving (1.2) has been considered including two linearly convergent algorithms for computing a dominant solvent [8,10].…”

Section: B Hashemi and M Dehghanmentioning

confidence: 99%

Efficient computation of enclosures for the exact solvents of a quadratic matrix equation

Hashemi

Dehghan

2010

ELA

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Abstract. None of the usual floating point numerical techniques available for solving the quadratic matrix equation AX 2 + BX + C = 0 with square matrices A, B, C and X, can provide an exact solution; they can just obtain approximations to an exact solution. We use interval arithmetic to compute an interval matrix which contains an exact solution to this quadratic matrix equation, where we aim at obtaining narrow intervals for each entry. We propose a residual version of a modified Krawczyk operator which has a cubic computational complexity, provided that A is nonsingular and X and X + A −1 B are diagonalizable. For the case that A is singular or nearly singular, but B is nonsingular we provide an enclosure method analogous to a functional iteration method. Numerical examples have also been given.

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Transforming algebraic Riccati equations into unilateral quadratic matrix equations

Cited by 46 publications

References 26 publications

A contour integral approach to the computation of invariant pairs

A contour integral approach to the computation of invariant pairs

Highly accurate doubling algorithms for M-matrix algebraic Riccati equations

Efficient computation of enclosures for the exact solvents of a quadratic matrix equation

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