Algorithms for Quadratic Matrix and Vector Equations

Poloni, Federico

doi:10.1007/978-88-7642-384-0

Cited by 8 publications

(8 citation statements)

References 116 publications

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“…Positivity of the solution has been proved for E3 in [11] in the case when M is irreducible. Earlier versions of the results appearing in this paper [26,27] contained an incorrect proof which failed to consider possible zero entries in P 0 e 0 .…”

Section: Concrete Casesmentioning

confidence: 93%

Quadratic vector equations

Poloni¹

2011

Algorithms for Quadratic Matrix and Vector Equations

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We study in a unified fashion several quadratic vector and matrix equations with nonnegativity hypotheses, by seeing them as special cases of the general problem M x = a + b (x, x), where a and the unknown x are componentwise nonnegative vectors, M is a nonsingular M-matrix, and b is a bilinear map from pairs of nonnegative vectors to nonnegative vectors. Specific cases of this equation have been studied extensively in the past by several authors, and include unilateral matrix equations from queuing problems [Bini, Latouche, Meini, 2005], nonsymmetric algebraic Riccati equations [Guo, Laub, 2000], and quadratic matrix equations encountered in neutron transport theory .We present a unified approach which treats the common aspects of their theoretical properties and basic iterative solution algorithms. This has interesting consequences: in some cases, we are able to derive in full generality theorems and proofs appeared in literature only for special cases of the problem; this broader view highlights the role of hypotheses such as the strict positivity of the minimal solution. In an example, we adapt an algorithm derived for one equation of the class to another, with computational advantage with respect to the existing methods. We discuss possible research lines, including the relationship among Newton-type methods and the cyclic reduction algorithm for unilateral quadratic equations.

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Section: Concrete Casesmentioning

confidence: 93%

Quadratic vector equations

Poloni¹

2011

Algorithms for Quadratic Matrix and Vector Equations

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“…The above compact form for determining E 0 , F 0 , G 0 , H 0 is based on a result of Poloni [19] (see also [4,Theorem 5.5]). It is easy (but tedious) to verify directly that these initial matrices are the same as those determined in [22].…”

Section: Numerical Methods For the Minimal Nonnegative Solutionmentioning

confidence: 99%

On algebraic Riccati equations associated with M -matrices

Guo

2013

Linear Algebra and its Applications

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We consider the algebraic Riccati equation for which the four coefficient matrices form an M -matrix K. When K is a nonsingular M -matrix or an irreducible singular M -matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M -matrix. Under a regularity assumption on the Mmatrix K, we show that the Riccati equation still has a minimal nonnegative solution. We also study the properties of this particular solution and explain how the solution can be found by existing methods.

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“…Eqn. ( 12) is a unimodular quadratic vector equation, which is the special case of vector equations of the form x + b( x, x) = c. This equation does not have an exact solution in closed form, but several numerical methods have been developed [29]. Here we use a heuristic method to identify what are the asymptotic values of W. Again we use the fact that in the limit t → ∞, we observe numerically the asymptotic values of W are either 0 or 1.…”

Section: Asymptotic State Recollection and Combinatorial Optimizationmentioning

confidence: 99%

Asymptotic Behavior of Memristive Circuits

Caravelli

2019

Entropy

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The interest in memristors has risen due to their possible application both as memory units and as computational devices in combination with CMOS. This is in part due to their nonlinear dynamics, and a strong dependence on the circuit topology. We provide evidence that also purely memristive circuits can be employed for computational purposes. In the present paper we show that a polynomial Lyapunov function in the memory parameters exists for the case of DC controlled memristors. Such Lyapunov function can be asymptotically approximated with binary variables, and mapped to quadratic combinatorial optimization problems. This also shows a direct parallel between memristive circuits and the Hopfield-Little model. In the case of Erdos-Renyi random circuits, we show numerically that the distribution of the matrix elements of the projectors can be roughly approximated with a Gaussian distribution, and that it scales with the inverse square root of the number of elements. This provides an approximated but direct connection with the physics of disordered system and, in particular, of mean field spin glasses. Using this and the fact that the interaction is controlled by a projector operator on the loop space of the circuit. We estimate the number of stationary points of the approximate Lyapunov function and provide a scaling formula as an upper bound in terms of the circuit topology only.

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Algorithms for Quadratic Matrix and Vector Equations

Cited by 8 publications

References 116 publications

Quadratic vector equations

Quadratic vector equations

On algebraic Riccati equations associated with M -matrices

Asymptotic Behavior of Memristive Circuits

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