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Academic literature on the topic 'Matrix Polynomial Problem'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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Journal articles on the topic "Matrix Polynomial Problem"

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Das, Biswajit, and Shreemayee Bora. "Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials." Electronic Journal of Linear Algebra 35 (February 1, 2019): 116–55. http://dx.doi.org/10.13001/1081-3810.3845.

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The complete eigenvalue problem associated with a rectangular matrix polynomial is typically solved via the technique of linearization. This work introduces the concept of generalized linearizations of rectangular matrix polynomials. For a given rectangular matrix polynomial, it also proposes vector spaces of rectangular matrix pencils with the property that almost every pencil is a generalized linearization of the matrix polynomial which can then be used to solve the complete eigenvalue problem associated with the polynomial. The properties of these vector spaces are similar to those introduced in the literature for square matrix polynomials and in fact coincide with them when the matrix polynomial is square. Further, almost every pencil in these spaces can be `trimmed' to form many smaller pencils that are strong linearizations of the matrix polynomial which readily yield solutions of the complete eigenvalue problem for the polynomial. These linearizations are easier to construct and are often smaller than the Fiedler linearizations introduced in the literature for rectangular matrix polynomials. Additionally, a global backward error analysis applied to these linearizations shows that they provide a wide choice of linearizations with respect to which the complete polynomial eigenvalue problem can be solved in a globally backward stable manner.

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陈, 贺. "Two Solutions of Matrix Polynomial Problem." Pure Mathematics 11, no. 02 (2021): 310–12. http://dx.doi.org/10.12677/pm.2021.112040.

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Dey, Papri. "Definite determinantal representations of multivariate polynomials." Journal of Algebra and Its Applications 19, no. 07 (July 23, 2019): 2050129. http://dx.doi.org/10.1142/s0219498820501297.

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In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial. Determinantal polynomials can characterize the feasible sets of semidefinite programming (SDP) problems that motivates us to deal with this problem. We introduce the notion of generalized mixed discriminant (GMD) of matrices which translates the determinantal representation problem into computing a point of a real variety of a specified ideal. We develop an algorithm to determine such a determinantal representation of a bivariate polynomial of degree [Formula: see text]. Then we propose a heuristic method to obtain a monic symmetric determinantal representation (MSDR) of a multivariate polynomial of degree [Formula: see text].

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Buyukkoroglu, Taner, Gokhan Celebi, and Vakif Dzhafarov. "On the Robust Stability of Polynomial Matrix Families." Electronic Journal of Linear Algebra 30 (February 8, 2015): 905–15. http://dx.doi.org/10.13001/1081-3810.3093.

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In this study, the problem of robust asymptotic stability of n by n polynomial matrix family, in both continuous-time and discrete-time cases, is considered. It is shown that in the continuous case the problem can be reduced to positivity of two specially constructed multivariable polynomials, whereas in the discrete-time case it is required three polynomials. A number of examples are given, where the Bernstein expansion method and sufficient conditions from [L.H. Keel and S.P. Bhattacharya. Robust stability via sign-definite decomposition. IEEE Transactions on Automatic Control, 56(1):140â145, 2011.] are applied to test positivity of the obtained multivariable polynomials. Sufficient conditions for matrix polytopes and one interesting negative result for companion matrices are also considered.

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Tansuwannont, Theerapat, Surachate Limkumnerd, Sujin Suwanna, and Pruet Kalasuwan. "Quantum Phase Estimation Algorithm for Finding Polynomial Roots." Open Physics 17, no. 1 (December 31, 2019): 839–49. http://dx.doi.org/10.1515/phys-2019-0087.

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AbstractQuantum algorithm is an algorithm for solving mathematical problems using quantum systems encoded as information, which is found to outperform classical algorithms in some specific cases. The objective of this study is to develop a quantum algorithm for finding the roots of nth degree polynomials where n is any positive integer. In classical algorithm, the resources required for solving this problem increase drastically when n increases and it would be impossible to practically solve the problem when n is large. It was found that any polynomial can be rearranged into a corresponding companion matrix, whose eigenvalues are roots of the polynomial. This leads to a possibility to perform a quantum algorithm where the number of computational resources increase as a polynomial of n. In this study, we construct a quantum circuit representing the companion matrix and use eigenvalue estimation technique to find roots of polynomial.

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Sharma, Bhuvnesh, Sunil Kumar, M. K. Paswan, and Dindayal Mahato. "Chebyshev Operational Matrix Method for Lane-Emden Problem." Nonlinear Engineering 8, no. 1 (January 28, 2019): 1–9. http://dx.doi.org/10.1515/nleng-2017-0157.

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AbstractIn the this paper, a new modified method is proposed for solving linear and nonlinear Lane-Emden type equations using first kind Chebyshev operational matrix of differentiation. The properties of first kind Chebyshev polynomial and their shifted polynomial are first presented. These properties together with the operation matrix of differentiation of first kind Chebyshev polynomial are utilized to obtain numerical solutions of a class of linear and nonlinear LaneEmden type singular initial value problems (IVPs). The absolute error of this method is graphically presented. The proposed framework is different from other numerical methods and can be used in differential equations of the same type. Several examples are illuminated to reveal the accuracy and validity of the proposed method.

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Grigoriu, Mircea. "Eigenvalue Problem for Uncertain Systems." Applied Mechanics Reviews 44, no. 11S (November 1, 1991): S89—S95. http://dx.doi.org/10.1115/1.3121377.

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Methods are developed for calculating probabilistic characteristics of the eigenvalues of stochastic symmetric matrices. The methods are based on the relationship between the elements of a matrix and its eigenvalues, perturbation method, bounds on eigenvalues, and zero-crossings of the characteristic polynomial. It is shown that the polynomial characteristic of a stochastic matrix can be viewed as a random process whose crossings of level zero define the eigenvalues of the matrix. The proposed methods of analysis are demonstrated by examples from dynamics and elasticity.

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GRIMBLE, M. J. "Polynomial matrix solution to the standardH2optimal control problem." International Journal of Systems Science 22, no. 5 (May 1991): 793–806. http://dx.doi.org/10.1080/00207729108910661.

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Djidjev, Hristo N., Georg Hahn, Susan M. Mniszewski, Christian F. A. Negre, and Anders M. N. Niklasson. "Using Graph Partitioning for Scalable Distributed Quantum Molecular Dynamics." Algorithms 12, no. 9 (September 7, 2019): 187. http://dx.doi.org/10.3390/a12090187.

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The simulation of the physical movement of multi-body systems at an atomistic level, with forces calculated from a quantum mechanical description of the electrons, motivates a graph partitioning problem studied in this article. Several advanced algorithms relying on evaluations of matrix polynomials have been published in the literature for such simulations. We aim to use a special type of graph partitioning to efficiently parallelize these computations. For this, we create a graph representing the zero–nonzero structure of a thresholded density matrix, and partition that graph into several components. Each separate submatrix (corresponding to each subgraph) is then substituted into the matrix polynomial, and the result for the full matrix polynomial is reassembled at the end from the individual polynomials. This paper starts by introducing a rigorous definition as well as a mathematical justification of this partitioning problem. We assess the performance of several methods to compute graph partitions with respect to both the quality of the partitioning and their runtime.

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HOANG, THANH MINH, and THOMAS THIERAUF. "ON THE MINIMAL POLYNOMIAL OF A MATRIX." International Journal of Foundations of Computer Science 15, no. 01 (February 2004): 89–105. http://dx.doi.org/10.1142/s0129054104002327.

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We investigate the complexity of the degree and the constant term of the minimal polynomial of a matrix. We show that the degree of the minimal polynomial is computationally equivalent to the matrix rank. We compare the constant term of the minimal polynomial with the constant term of the characteristic polynomial. The latter is known to be computable in the logspace counting class GapL. We show that if this holds for the minimal polynomial as well, then the exact counting in logspace class C=L is closed under complement. Whether C=L is closed under complement is one of the main open problems in this area. As an application of our techniques we show that the problem of deciding whether a matrix is diagonalizable is complete for AC0(C=L), the AC0-closure ofC=L.

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Dissertations / Theses on the topic "Matrix Polynomial Problem"

1

Wang, Ting. "Algorithms for parallel and sequential matrix-chain product problem." Ohio : Ohio University, 1997. http://www.ohiolink.edu/etd/view.cgi?ohiou1184355429.

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Koller, Angela Erika. "The frequency assignment problem." Thesis, Brunel University, 2004. http://bura.brunel.ac.uk/handle/2438/4967.

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This thesis examines a wide collection of frequency assignment problems. One of the largest topics in this thesis is that of L(2,1)-labellings of outerplanar graphs. The main result in this topic is the fact that there exists a polynomial time algorithm to determine the minimum L(2,1)-span for an outerplanar graph. This result generalises the analogous result for trees, solves a stated open problem and complements the fact that the problem is NP-complete for planar graphs. We furthermore give best possible bounds on the minimum L(2,1)-span and the cyclic-L(2,1)-span in outerplanar graphs, when the maximum degree is at least eight. We also give polynomial time algorithms for solving the standard constraint matrix problem for several classes of graphs, such as chains of triangles, the wheel and a larger class of graphs containing the wheel. We furthermore introduce the concept of one-close-neighbour problems, which have some practical applications. We prove optimal results for bipartite graphs, odd cycles and complete multipartite graphs. Finally we evaluate different algorithms for the frequency assignment problem, using domination analysis. We compute bounds for the domination number of some heuristics for both the fixed spectrum version of the frequency assignment problem and the minimum span frequency assignment problem. Our results show that the standard greedy algorithm does not perform well, compared to some slightly more advanced algorithms, which is what we would expect. In this thesis we furthermore give some background and motivation for the topics being investigated, as well as mentioning several open problems.

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Potts, Daniel, and Manfred Tasche. "Parameter estimation for nonincreasing exponential sums by Prony-like methods." Universitätsbibliothek Chemnitz, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-86476.

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For noiseless sampled data, we describe the close connections between Prony--like methods, namely the classical Prony method, the matrix pencil method and the ESPRIT method. Further we present a new efficient algorithm of matrix pencil factorization based on QR decomposition of a rectangular Hankel matrix. The algorithms of parameter estimation are also applied to sparse Fourier approximation and nonlinear approximation.

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Ali, Ali Hasan. "Modifying Some Iterative Methods for Solving Quadratic Eigenvalue Problems." Wright State University / OhioLINK, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=wright1515029541712239.

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Sharify, Meisam. "Algorithmes de mise à l'échelle et méthodes tropicales en analyse numérique matricielle." Phd thesis, Ecole Polytechnique X, 2011. http://pastel.archives-ouvertes.fr/pastel-00643836.

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L'Algèbre tropicale peut être considérée comme un domaine relativement nouveau en mathématiques. Elle apparait dans plusieurs domaines telles que l'optimisation, la synchronisation de la production et du transport, les systèmes à événements discrets, le contrôle optimal, la recherche opérationnelle, etc. La première partie de ce manuscrit est consacrée a l'étude des applications de l'algèbre tropicale à l'analyse numérique matricielle. Nous considérons tout d'abord le problème classique de l'estimation des racines d'un polynôme univarié. Nous prouvons plusieurs nouvelles bornes pour la valeur absolue des racines d'un polynôme en exploitant les méthodes tropicales. Ces résultats sont particulièrement utiles lorsque l'on considère des polynômes dont les coefficients ont des ordres de grandeur différents. Nous examinons ensuite le problème du calcul des valeurs propres d'une matrice polynomiale. Ici, nous introduisons une technique de mise à l'échelle générale, basée sur l'algèbre tropicale, qui s'applique en particulier à la forme compagnon. Cette mise à l'échelle est basée sur la construction d'une fonction polynomiale tropicale auxiliaire, ne dépendant que de la norme des matrices. Les raciness (les points de non-différentiabilité) de ce polynôme tropical fournissent une pré-estimation de la valeur absolue des valeurs propres. Ceci se justifie en particulier par un nouveau résultat montrant que sous certaines hypothèses faites sur le conditionnement, il existe un groupe de valeurs propres bornées en norme. L'ordre de grandeur de ces bornes est fourni par la plus grande racine du polynôme tropical auxiliaire. Un résultat similaire est valable pour un groupe de petites valeurs propres. Nous montrons expérimentalement que cette mise à l'échelle améliore la stabilité numérique, en particulier dans des situations où les données ont des ordres de grandeur différents. Nous étudions également le problème du calcul des valeurs propres tropicales (les points de non-différentiabilité du polynôme caractéristique) d'une matrice polynômiale tropicale. Du point de vue combinatoire, ce problème est équivalent à trouver une fonction de couplage: la valeur d'un couplage de poids maximum dans un graphe biparti dont les arcs sont valués par des fonctions convexes et linéaires par morceaux. Nous avons développé un algorithme qui calcule ces valeurs propres tropicales en temps polynomial. Dans la deuxième partie de cette thèse, nous nous intéressons à la résolution de problèmes d'affectation optimale de très grande taille, pour lesquels les algorithms séquentiels classiques ne sont pas efficaces. Nous proposons une nouvelle approche qui exploite le lien entre le problème d'affectation optimale et le problème de maximisation d'entropie. Cette approche conduit à un algorithme de prétraitement pour le problème d'affectation optimale qui est basé sur une méthode itérative qui élimine les entrées n'appartenant pas à une affectation optimale. Nous considérons deux variantes itératives de l'algorithme de prétraitement, l'une utilise la méthode Sinkhorn et l'autre utilise la méthode de Newton. Cet algorithme de prétraitement ramène le problème initial à un problème beaucoup plus petit en termes de besoins en mémoire. Nous introduisons également une nouvelle méthode itérative basée sur une modification de l'algorithme Sinkhorn, dans lequel un paramètre de déformation est lentement augmenté. Nous prouvons que cette méthode itérative(itération de Sinkhorn déformée) converge vers une matrice dont les entrées non nulles sont exactement celles qui appartiennent aux permutations optimales. Une estimation du taux de convergence est également présentée.

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Munro, Christopher James. "Algorithms for matrix polynomials and structured matrix problems." Thesis, University of Manchester, 2011. https://www.research.manchester.ac.uk/portal/en/theses/algorithms-for-matrix-polynomials-and-structured-matrix-problems(9154f9f0-8b86-46f8-8066-40c5139fcc51).html.

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Murfitt, Louise. "Discrete event dynamic systems in max-algebra : realisation and related combinatorial problems." Thesis, University of Birmingham, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.368451.

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García-García, David. "Schur Averages in Random Matrix Ensembles." Doctoral thesis, 2019. http://hdl.handle.net/10451/45592.

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The main focus of this PhD thesis is the study of minors of Toeplitz, Hankel and Toeplitz±Hankel matrices. These can be expressed as matrix models over the classical Lie groups G(N) = U(N); Sp(2N);O(2N);O(2N + 1), with the insertion of irreducible characters associated to each of the groups. In order to approach this topic, we consider matrices generated by formal power series in terms of symmetric functions. We exploit these connections to obtain several relations between the models over the different groups G(N), and to investigate some of their structural properties. We compute explicitly several objects of interest, including a variety of matrix models, evaluations of certain skew Schur polynomials, partition functions and Wilson loops of G(N) Chern-Simons theory on S3, and fermion quantum models with matrix degrees of freedom. We also explore the connection with orthogonal polynomials, and study the large N behaviour of the average of a characteristic polynomial in the Laguerre Unitary Ensemble by means of the associated Riemann-Hilbert problem. We gratefully acknowledge the support of the Fundação para a Ciência e a Tecnologia through its LisMath scholarship PD/BD/113627/2015, which made this work possible.

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Zhan, Xuzhou. "On matrix generalization of Hurwitz polynomials." Doctoral thesis, 2017. https://ul.qucosa.de/id/qucosa%3A16415.

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This thesis focuses on matrix generalizations of Hurwitz polynomials. A real polynomial with all its roots in the open left half plane of the complex plane is called a Hurwitz polynomial. The study of these Hurwitz polynomials has a long and abundant history, which is associated with the names of Hermite, Routh, Hurwitz, Liénard, Chipart, Wall, Gantmacher et al. The direct matricial generalization of Hurwitz polynomials is naturally defined as follows: A p by p matrix polynomial F is called a Hurwitz matrix polynomial if the determinant of F is a Hurwitz polynomial. Recently, Choque Rivero followed another line of matricial extensions of the classical Hurwitz polynomial, called matrix Hurwitz type polynomials. However, the notion “matrix Hurwitz type polynomial” is still irrelative to “Hurwitz matrix polynomial” due to the totally unclear zero location of the former notion. So the main goal of this thesis is to discover the relation between the two notions “matrix Hurwitz-type polynomials” and “Hurwitz matrix polynomials' and provide some criteria to identify Hurwitz matrix polynomials. The central idea is to determine the inertia triple of matrix polynomials in terms of some related matrix sequences. Suppose that F is a p by p matrix-valued polynomial of degree n. We split F into the odd part and the even part, which allow us to introduce an essential rational matrix function of right type G. From the matrix coefficients of the Laurent series of G we construct the (n-1)-th extended sequence of right Markov parameters (SRMP) of F. Then we show that the inertia triple of F can be characterized by a combination of the inertia triples of two block Hankel matrices generated by the (n-1)-th SRMP of F and the number of zeros (counting for multiplicities) of greatest right common divisors of the even part and the odd part of F lying on the left half of the real axis. By an analogous approach we also obtain the dual results for the inertia triple of F in terms of the SLMP of F. Then we demonstrate that F is a Hurwitz matrix polynomial of degree n if and only if the (n − 1)-th SRMP (resp. SLMP) of F is a Stieltjes positive definite sequence. On this account, the two notions “Hurwitz matrix polynomials” and “matrix Hurwitz type polynomials” are equivalent. In addition, we investigate quasi-stable matrix polynomials appearing in the theory of stability, which contain Hurwitz matrix polynomials as a special case. We seek a correspondence between quasi-stable matrix polynomials, Stieltjes moment problems and multiple Nevanlinna-Pick interpolation in the Stieltjes class. Accordingly, we prove that F is a quasi-stable matrix polynomial if and only if the (n − 1)-th SRMP (resp. SLMP) of F is a Stieltjes non-negative definite extendable sequence and the zeros of right (resp. left) greatest common divisors of the even part and the odd part of F are located on the left half of the real axis.:1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Matrix polynomials and greatest common divisors. . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Greatest common divisors of matrix polynomials . . . . . . . . . . . . . . . . . . . . . 8 3 Matrix sequences and their connection to truncated matricial moment problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Matrix fraction description and some related topics . . . . . . . . . . . . . . . . . . 19 4.1 Realization of Matrix fraction description from Markov parameters . . . . . . . 19 4.2 The interrelation between Hermitian transfer function matrices and monic orthogonal system of matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . .27 5 The Bezoutian of matrix polynomials and the inertia problem of matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 5.2 The Anderson-Jury Bezoutian matrices in connection to special transfer function matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 6 Para-Hermitian strictly proper transfer function matrices and their related monic Hurwitz matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7 Solution of matricial Routh-Hurwitz problems in terms of the Markov pa- rameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 8 Matrix Hurwitz type polynomials and some related topics . . . . . . . . . . . . . . 67 9 Hurwitz matrix polynomials and some related topics . . . . . . . . . . . . . . . . . . 77 9.1 Hurwitz matrix polynomials, Stieltjes positive definite sequences and matrix Hurwitz type polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 9.2 S -system of Hurwitz matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 82 10 Quasi-stable matrix polynomials and some related topics . . . . . . . . . . . . 95 10.1 Particular monic quasi-stable matrix polynomials and Stieltjes moment problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95 10.2 Particular monic quasi-stable matrix polynomials and multiple Nevanlinna- Pick interpolation in the Stieltjes class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101 10.3 General description of monic quasi-stable matrix polynomials . . . . . . . . .104 List of terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 List of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Selbständigkeitserklärung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

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Teng, Chia-Chun, and 鄧家駿. "Nonequivalence transformations for matrix polynomial eigenvalue problems in model tuning." Thesis, 1999. http://ndltd.ncl.edu.tw/handle/39283481455176448510.

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碩士
國立清華大學
數學系
87
In this thesis, our motivation is solving a practical example of special form A. A method with the nonequivalence transformation is presented for eigenvalue embedding from the identified system. The method is given for the special form of A in the state-space model of an aircraft in Boeing. We review two identified methods and present an algorithm of the method with nonequivalence transformations. The numerical result indicates that the algorithm would keep the proper form of A and show the differences between the original matrices of the P.D.E terms and the transformed ones. Then we can investigate what is wrong with the aircraft.

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Books on the topic "Matrix Polynomial Problem"

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International Conference on p-Adic Functional Analysis (11th 2010 Université Blaise Pascal). Advances in non-Archimedean analysis: Eleventh International Conference on p-Adic Functional Analysis, July 5-9 2010, Université Blaise Pascal, Clermont-Ferrand, France. Edited by Araujo-Gomez Jesus 1965-, Diarra B. (Bertin) 1944-, and Escassut Alain. Providence, R.I: American Mathematical Society, 2011.

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Olshanski, Grigori. Enumeration of maps. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.26.

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This article discusses the relationship between random matrices and maps, i.e. graphs drawn on surfaces, with particular emphasis on the one-matrix model and how it can be used to solve a map enumeration problem. It first provides an overview of maps and related objects, recalling the basic definitions related to graphs and defining maps as graphs embedded into surfaces before considering a coding of maps by pairs of permutations. It then examines the connection between matrix integrals and maps, focusing on the Hermitian one-matrix model with a polynomial potential and how the formal expansion of its free energy around a Gaussian point (quadratic potential) can be represented by diagrams identifiable with maps. The article also illustrates how the solution of the map enumeration problem can be deduced by means of random matrix theory (RMT). Finally, it explains how the matrix model result can be translated into a bijective proof.

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Akemann, Gernot. Random matrix theory and quantum chromodynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797319.003.0005.

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This chapter was originally presented to a mixed audience of physicists and mathematicians with some basic working knowledge of random matrix theory. The first part is devoted to the solution of the chiral Gaussian unitary ensemble in the presence of characteristic polynomials, using orthogonal polynomial techniques. This includes all eigenvalue density correlation functions, smallest eigenvalue distributions, and their microscopic limit at the origin. These quantities are relevant for the description of the Dirac operator spectrum in quantum chromodynamics with three colors in four Euclidean space-time dimensions. In the second part these two theories are related based on symmetries, and the random matrix approximation is explained. In the last part recent developments are covered, including the effect of finite chemical potential and finite space-time lattice spacing, and their corresponding orthogonal polynomials. This chapter also provides some open random matrix problems.

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Center, Langley Research, ed. On the numerical formulation of parametric linear fractional transformation (LFT) uncertainty models for multivariate matrix polynomial problems. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1998.

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Book chapters on the topic "Matrix Polynomial Problem"

1

Ball, Joseph A., Israel Gohberg, and Leiba Rodman. "Polynomial Interpolation Problems Based on Divisibility." In Interpolation of Rational Matrix Functions, 213–23. Basel: Birkhäuser Basel, 1990. http://dx.doi.org/10.1007/978-3-0348-7709-1_11.

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Branquinho, Amílcar, Ana Foulquié-Moreno, and Manuel Mañas-Baena. "Riemann–Hilbert Problem and Matrix Biorthogonal Polynomials." In Orthogonal Polynomials: Current Trends and Applications, 1–15. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-56190-1_1.

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Ball, Joseph A., Israel Gohberg, and Leiba Rodman. "Null Structure and Interpolation Problems for Matrix Polynomials." In Interpolation of Rational Matrix Functions, 40–64. Basel: Birkhäuser Basel, 1990. http://dx.doi.org/10.1007/978-3-0348-7709-1_3.

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Ball, Joseph A., Israel Gohberg, and Leiba Rodman. "Interpolation Problems for Matrix Polynomials and Rational Matrix-Functions." In Interpolation of Rational Matrix Functions, 317–70. Basel: Birkhäuser Basel, 1990. http://dx.doi.org/10.1007/978-3-0348-7709-1_17.

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Mackey, D. Steven, Niloufer Mackey, and Françoise Tisseur. "Polynomial Eigenvalue Problems: Theory, Computation, and Structure." In Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory, 319–48. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-15260-8_12.

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Azizov, T. Ya, A. Dijksma, K. H. Förster, and P. Jonas. "Quadratic (Weakly) Hyperbolic Matrix Polynomials: Direct and Inverse Spectral Problems." In Recent Advances in Operator Theory in Hilbert and Krein Spaces, 11–40. Basel: Birkhäuser Basel, 2009. http://dx.doi.org/10.1007/978-3-0346-0180-1_2.

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Ellis, Robert L., Israel Gohberg, and David C. Lay. "Distribution of Zeros of Matrix-Valued Continuous Analogues of Orthogonal Polynomials." In Continuous and Discrete Fourier Transforms, Extension Problems and Wiener-Hopf Equations, 26–70. Basel: Birkhäuser Basel, 1992. http://dx.doi.org/10.1007/978-3-0348-8596-6_2.

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Gu, Shenshen, Jiao Peng, and Rui Cui. "A Polynomial Time Solvable Algorithm to Binary Quadratic Programming Problems with Q Being a Seven-Diagonal Matrix and Its Neural Network Implementation." In Advances in Neural Networks – ISNN 2014, 338–46. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-12436-0_38.

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"Chapter 5: Generalized and Matrix Polynomial Eigenvalue Problems." In Core-Chasing Algorithms for the Eigenvalue Problem, 89–104. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2018. http://dx.doi.org/10.1137/1.9781611975345.ch5.

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Fallat, Shaun M., and Charles R. Johnson. "Recognition." In Totally Nonnegative Matrices. Princeton University Press, 2011. http://dx.doi.org/10.23943/princeton/9780691121574.003.0004.

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This chapter discusses the recognition of TN matrices. It touches on one of the many applications for the structure of TN matrices. TN matrices enjoy tremendous structure, as a result of requiring all minors to be nonnegative. This intricate structure makes it easier to determine when a matrix is TP than to check when it is a P-matrix, which formally involves far fewer minors. Vandermonde matrices arise in the problem of determining a polynomial of degree at most n − 1 that interpolates n data points. Suppose that n data points (xᵢ,yᵢ)unconverted formula are given. The goal is to construct a polynomial p(x) = a₀ + a₁x + … + asubscript n − 1xsuperscript n − 1 that satisfies p(xᵢ) = yᵢ for i = 1, 2, …,n.

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Conference papers on the topic "Matrix Polynomial Problem"

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Chen, Been-Der, and Sanjay Lall. "Degree Bounds for Polynomial Verification of the Matrix Cube Problem." In Proceedings of the 45th IEEE Conference on Decision and Control. IEEE, 2006. http://dx.doi.org/10.1109/cdc.2006.376783.

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Sunkari, Rajesh Pavan, and Linda C. Schmidt. "Laplace and Extended Adjacency Matrices for Isomorphism Detection of Kinematic Chains Using the Characteristic Polynomial Approach." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84609.

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The kinematic chain isomorphism problem is one of the most challenging problems facing mechanism researchers. Methods using the spectral properties, characteristic polynomial and eigenvectors, of the graph related matrices were developed in literature for isomorphism detection. Detection of isomorphism using only the spectral properties corresponds to a polynomial time isomorphism detection algorithm. However, most of the methods used are either computationally inefficient or unreliable (i.e., failing to identify non-isomorphic chains). This work establishes the reliability of using the characteristic polynomial of the Laplace matrix for isomorphism detection of a kinematic chain. The Laplace matrix of a graph is used extensively in the field of algebraic graph theory for characterizing a graph using its spectral properties. The reliability in isomorphism detection of the characteristic polynomial of the Laplace matrix was comparable with that of the adjacency matrix. However, using the characteristic polynomials of both the matrices is superior to using either method alone. In search for a single matrix whose characteristic polynomial unfailingly detects isomorphism, novel matrices called the extended adjacency matrices are developed. The reliability of the characteristic polynomials of these matrices is established. One of the proposed extended adjacency matrices is shown to be the best graph matrix for isomorphism detection using the characteristic polynomial approach.

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Grimble, M. J. "Discrete polynomial matrix solution of the standard state-feedback H∞ control problem." In 1993 American Control Conference. IEEE, 1993. http://dx.doi.org/10.23919/acc.1993.4793030.

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Zheng, Qian, and Fen Wu. "State Feedback and Output Feedback Control of Polynomial Nonlinear Systems Using Fractional Lyapunov Functions." In ASME 2007 International Mechanical Engineering Congress and Exposition. ASMEDC, 2007. http://dx.doi.org/10.1115/imece2007-42147.

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In this paper, we will study the state feedback control problem of polynomial nonlinear systems using fractional Lyapunov functions. By adding constraints to bound the variation rate of each state, the general difficulty of calculating derivative of nonquadratic Lyapunov function is effectively overcome. As a result, the state feedback conditions are simplified as a set of Linear Matrix Inequalities (LMIs) with polynomial entries. Computationally tractable solution is obtained by Sum-of-Squares (SOS) decomposition. And it turns out that both of the Lyapunov matrix and the state feedback gain are state dependent fractional matrix functions, where the numerator as well as the denominator can be polynomials with flexible forms and higher nonlinearities involved in. Same idea is extended to a class of output dependent nonlinear systems and the stabilizing output feedback controller is specified as polynomial of output. Synthesis conditions are similarly derived as using constant Lyapunov function except that all entries in LMIs are polynomials of output with derivative of output involved in. By bounding the variation rate of output and gridding on the bounded interval, the LMIs are solvable by SOS decomposition. Finally, two examples are used to materialize the design scheme and clarify the various choices on state boundaries.

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Ren, Ping, and Clément Gosselin. "Trajectory Planning of Cable-Suspended Parallel Robots Using Interval Positive-Definite Polynomials." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-71205.

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In this paper, the dynamic point-to-point trajectory planning of cable-suspended robots is investigated. A simple planar two-degree-of-freedom (2-dof) robot is used to demonstrate the technique. In order to maintain the cables’ positive tension, a set of algebraic inequalities is derived from the dynamic model of the 2-dof robot. The trajectories are defined using parametric polynomials with the coefficients determined by the prescribed initial and final states, and the variable time duration. With the polynomials substituted into the inequality constraints, the planning problem is then converted into an algebraic investigation on how the coefficients of the polynomials will affect the number of real roots over a given interval. An analytical approach based on a polynomial’s Discrimination Matrix and Discriminant Sequence is proposed to solve the problem. It is shown that, by adjusting the time duration within appropriate ranges, it is possible to find positive-definite polynomials such that the polynomial-based trajectories always satisfy the inequality constraints of the dynamic system. Feasible dynamic trajectories that are able to travel both beyond and within the static workspace will exploit more potential of the cable-suspended robotic platform.

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Dhingra, A. K., A. N. Almadi, and D. Kohli. "A Gröbner-Sylvester Hybrid Method for Closed-Form Displacement Analysis of Mechanisms." In ASME 1998 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1998. http://dx.doi.org/10.1115/detc98/mech-5969.

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Abstract The displacement analysis problem for planar and spatial mechanisms can be written as a system of multivariate polynomial equations. Elimination theory based on resultants and polynomial continuation are some of the methods which have been used to solve this problem. This paper presents a new approach to displacement analysis using the reduced Gröbner basis form of a system of equations under degree lexicographic (dlex) term ordering of its monomials and Sylvester’s Dialytic elimination method. Using the Gröbner-Sylvester hybrid approach, a finitely solvable system of equations F is transformed into its reduced Gröbner basis G using dlex term ordering. Next, using the entire or a subset of the set of generators in G, the Sylvester’s matrix is assembled. The vanishing of the resultant, given as the determinant of Sylvester’s matrix, yields the necessary and sufficient condition for the polynomials in G (as well as F) to have a common factor. The proposed approach appears to provide a systematic and rational procedure to the problem discussed by Roth (1994) dealing with the generation of (additional) equations for constructing the Sylvester’s matrix. Three examples illustrating the applicability of the proposed approach to displacement analysis of planar and spatial mechanisms are presented. The first and second examples deal with forward displacement analysis of the general 6-6 Stewart mechanism and the 6-6 Stewart platform, whereas the third example deals with the determination of the input-output polynomial of a 8-link 1-DOF mechanism which does not contain any 4-link loops.

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Shao, Yan-Lin, and Odd M. Faltinsen. "Towards Efficient Fully-Nonlinear Potential-Flow Solvers in Marine Hydrodynamics." In ASME 2012 31st International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/omae2012-83319.

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Solving potential-flow problems using the Boundary Element Method (BEM) is a strong tradition in marine hydrodynamics. An early example of the application of BEM is by Bai & Yeung [1]. The bottleneck of the conventional BEM in terms of CPU time and computer memory arises as the number of unknowns increases. Wu & Eatock Taylor [2] suggested that the Finite Element Method (FEM) field solver is much faster than the BEM based on their comparisons in a wave making problem. In this paper, we aim to find a highly efficient method to solve fully-nonlinear wave-body interaction problems based on potential-flow theory. We compare the efficiency and the accuracy of five different methods for the potential flows in two dimensions (2D), two of which are BEM-based while the other three are field solvers. The comparisons indicate that it is beneficial to use either an accelerated matrix-free BEM, e.g. Fast Multipole Method accelerated BEM (FMM-BEM), or any field solvers whose resulting matrix are sparse. Another highlight of this paper is that an efficient numerical potential-flow method named the harmonic polynomial cell (HPC) method is developed. The flow in each cell is described by a set of harmonic polynomials. The presented procedure has approximately 4th order accuracy, while its resulting matrix is sparse similarly as the other field solvers, e.g. Finite Element Method (FEM), Finite Difference Method (FDM) and Finite Volume Method (FVM). The method is verified by a linear wave making problem for which the steady-state analytical solution is available, and the forced oscillation of a semi-submerged circular cylinder for which the frequency-domain added mass and damping coefficients are compared. The fully-nonlinear wave making problem and nonlinear propagating waves over a submerged bar are also studied for validation purposes. Only 2D cases are studied in this paper.

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Chen, Zhihuai, Yinan Li, Xiaoming Sun, Pei Yuan, and Jialin Zhang. "A Quantum-inspired Classical Algorithm for Separable Non-negative Matrix Factorization." In Twenty-Eighth International Joint Conference on Artificial Intelligence {IJCAI-19}. California: International Joint Conferences on Artificial Intelligence Organization, 2019. http://dx.doi.org/10.24963/ijcai.2019/627.

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Non-negative Matrix Factorization (NMF) asks to decompose a (entry-wise) non-negative matrix into the product of two smaller-sized nonnegative matrices, which has been shown intractable in general. In order to overcome this issue, separability assumption is introduced which assumes all data points are in a conical hull. This assumption makes NMF tractable and widely used in text analysis and image processing, but still impractical for huge-scale datasets. In this paper, inspired by recent development on dequantizing techniques, we propose a new classical algorithm for separable NMF problem. Our new algorithm runs in polynomial time in the rank and logarithmic in the size of input matrices, which achieves an exponential speedup in the low-rank setting.

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Rai, K. N., and D. C. Rai. "A Finite Element Method for the Solution of Free Boundary Problem." In ASME 2004 Heat Transfer/Fluids Engineering Summer Conference. ASMEDC, 2004. http://dx.doi.org/10.1115/ht-fed2004-56777.

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A finite element method is presented for the solution of a free boundary problem which arises during planar melting of a semi-infinite medium initially at a temperature which is slightly below the melting temperature of the solid. The surface temperature is assumed to vary with time. Two different situations are considered (I) when thermal diffusivity is independent of temperature and (II) when thermal diffusivity varies linearly with temperature. The differential equation governing the process is converted to initial value problem of vector matrix form. The time function is approximated by Chebyshev series and the operational matrix of integration is applied, a linear differential equation can be represented by a set of linear algebraic equations and a nonlinear differential equation can be represented by a set of nonlinear algebraic equations. The solution of the problem is then found in terms of Chebyshev polynomial of second kind. The solution of this initial value problem is utilized iteratively in the interface heat flux equation to determine interface location as well as the temperature in two regions. The method appears to be accurate in cases for which closed form solutions are available, it agrees well with them. The effect of several parameters on the melting are analysed and discussed.

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Kolansky, Jeremy, and Corina Sandu. "Generalized Polynomial Chaos-Based Extended Kalman Filter: Improvement and Expansion." In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/detc2013-12082.

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The generalized polynomial chaos (gPC) method for propagating uncertain parameters through dynamical systems (previously developed at Virginia Tech) has been shown to be very computationally efficient. This method seems also to be ideal for real-time parameter estimation when merged with the Extended Kalman Filter (EKF). The resulting technique is shown in the present paper for systems in state-space representations, and then expanded to systems in regressions formulations. Due to the way the filter interacts with the polynomial chaos expansions, the covariance matrix is forced to zero in finite time. This problem shows itself as an inability to perform state estimations and causes the parameters to converge to incorrect values for state space systems. In order to address this issue, improvements to the method are implemented and the updated method is applied to both state space and regression systems. The resultant technique shows high accuracy of both state and parameter estimations.

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