Journal articles: '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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Yüzbaşı, Şuayip. "A Laguerre Approach for the Solutions of Singular Perturbated Differential Equations." International Journal of Computational Methods 14, no. 04 (April 18, 2017): 1750034. http://dx.doi.org/10.1142/s0219876217500347.

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In this paper, a Laguerre method is presented to solve singularly perturbated two-point boundary value problems. By means of the matrix relations of the Laguerre polynomials and their derivatives, original problem is transformed into a matrix equation. Later, we use collocation points in the matrix equation and thus the considered problem is reduced to a system of linear algebraic equations. The solution of this system gives the coefficients of the desired approximate solution. Also, an error estimation based on the residual function is introduced for the method. The Laguerre polynomial solution is improved by using this error estimation. Finally, error estimation and residual improvement are illustrated by examples and comparisons are given with other methods.

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Guersenzvaig, N. H., and Fernando Szechtman. "Generalized Artin–Schreier polynomials." Journal of Algebra and Its Applications 14, no. 06 (April 21, 2015): 1550084. http://dx.doi.org/10.1142/s021949881550084x.

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Let F be a field of prime characteristic p containing 𝔽pn as a subfield. We refer to q(X) = Xpn - X - a ∈ F[X] as a generalized Artin–Schreier polynomial. Suppose that q(X) is irreducible and let Cq(X) be the companion matrix of q(X). Then ad Cq(X) has such highly unusual properties that any A ∈ 𝔤𝔩(m) such that ad A has like properties is shown to be similar to the companion matrix of an irreducible generalized Artin–Schreier polynomial. We discuss close connections with the decomposition problem of the tensor product of indecomposable modules for a one-dimensional Lie algebra over a field of characteristic p, the problem of finding an explicit primitive element for every intermediate field of the Galois extension associated to an irreducible generalized Artin–Schreier polynomial, and the problem of finding necessary and sufficient conditions for the irreducibility of a family of polynomials.

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Karampetakis, N. P. "The output zeroing problem for general polynomial matrix descriptions." International Journal of Control 71, no. 6 (January 1998): 1069–86. http://dx.doi.org/10.1080/002071798221489.

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Jiang, Zhaolin. "Fast Algorithms for Solving FLSR-Factor Block Circulant Linear Systems and Inverse Problem ofAX=b." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/340803.

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Block circulant and circulant matrices have already become an ideal research area for solving various differential equations. In this paper, we give the definition and the basic properties of FLSR-factor block circulant (retrocirculant) matrix over fieldF. Fast algorithms for solving systems of linear equations involving these matrices are presented by the fast algorithm for computing matrix polynomials. The unique solution is obtained when such matrix over a fieldFis nonsingular. Fast algorithms for solving the unique solution of the inverse problem ofAX=bin the class of the level-2 FLS(R,r)-circulant(retrocirculant) matrix of type(m,n)over fieldFare given by the right largest common factor of the matrix polynomial. Numerical examples show the effectiveness of the algorithms.

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Chitambar, Eric, Carl Miller, and Yaoyun Shi. "Deciding unitary equivalence between matrix polynomials and sets of bipartite quantum states." Quantum Information and Computation 11, no. 9&10 (September 2011): 813–19. http://dx.doi.org/10.26421/qic11.9-10-6.

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In this brief report, we consider the equivalence between two sets of $m+1$ bipartite quantum states under local unitary transformations. For pure states, this problem corresponds to the matrix algebra question of whether two degree $m$ matrix polynomials are unitarily equivalent; i.e. $UA_iV^\dagger=B_i$ for $0\leq i\leq m$ where $U$ and $V$ are unitary and $(A_i, B_i)$ are arbitrary pairs of rectangular matrices. We present a randomized polynomial-time algorithm that solves this problem with an arbitrarily high success probability and outputs transforming matrices $U$ and $V$.

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Shavarovskii, B. Z. "Toeplitz Matrices in the Problem of Semiscalar Equivalence of Second-Order Polynomial Matrices." International Journal of Analysis 2017 (October 26, 2017): 1–14. http://dx.doi.org/10.1155/2017/6701078.

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We consider the problem of determining whether two polynomial matrices can be transformed to one another by left multiplying with some nonsingular numerical matrix and right multiplying by some invertible polynomial matrix. Thus the equivalence relation arises. This equivalence relation is known as semiscalar equivalence. Large difficulties in this problem arise already for 2-by-2 matrices. In this paper the semiscalar equivalence of polynomial matrices of second order is investigated. In particular, necessary and sufficient conditions are found for two matrices of second order being semiscalarly equivalent. The main result is stated in terms of determinants of Toeplitz matrices.

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Deng, Quanling. "Analytical solutions to some generalized and polynomial eigenvalue problems." Special Matrices 9, no. 1 (January 1, 2021): 240–56. http://dx.doi.org/10.1515/spma-2020-0135.

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Abstract It is well-known that the finite difference discretization of the Laplacian eigenvalue problem −Δu = λu leads to a matrix eigenvalue problem (EVP) Ax =λx where the matrix A is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in a recent work of Strang and MacNamara. We generalize the results and develop analytical solutions to certain generalized matrix eigenvalue problems (GEVPs) Ax = λBx which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation. Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities.

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Liu, Shunyi. "Generalized Permanental Polynomials of Graphs." Symmetry 11, no. 2 (February 16, 2019): 242. http://dx.doi.org/10.3390/sym11020242.

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The search for complete graph invariants is an important problem in graph theory and computer science. Two networks with a different structure can be distinguished from each other by complete graph invariants. In order to find a complete graph invariant, we introduce the generalized permanental polynomials of graphs. Let G be a graph with adjacency matrix A ( G ) and degree matrix D ( G ) . The generalized permanental polynomial of G is defined by P G ( x , μ ) = per ( x I − ( A ( G ) − μ D ( G ) ) ) . In this paper, we compute the generalized permanental polynomials for all graphs on at most 10 vertices, and we count the numbers of such graphs for which there is another graph with the same generalized permanental polynomial. The present data show that the generalized permanental polynomial is quite efficient for distinguishing graphs. Furthermore, we can write P G ( x , μ ) in the coefficient form ∑ i = 0 n c μ i ( G ) x n − i and obtain the combinatorial expressions for the first five coefficients c μ i ( G ) ( i = 0 , 1 , ⋯ , 4 ) of P G ( x , μ ) .

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Rakhshan, Mohsen, Navid Vafamand, Mohammad Mehdi Mardani, Mohammad-Hassan Khooban, and Tomislav Dragičević. "Polynomial control design for polynomial systems: A non-iterative sum of squares approach." Transactions of the Institute of Measurement and Control 41, no. 7 (October 1, 2018): 1993–2004. http://dx.doi.org/10.1177/0142331218793476.

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This paper proposes a non-iterative state feedback design approach for polynomial systems using polynomial Lyapunov function based on the sum of squares (SOS) decomposition. The polynomial Lyapunov matrix consists of states of the system leading to the non-convex problem. A lower bound on the time derivative of the Lyapunov matrix is considered to turn the non-convex problem into a convex one; and hence, the solutions are computed through semi-definite programming methods in a non-iterative fashion. Furthermore, we show that the proposed approach can be applied to a wide range of practical and industrial systems that their controller design is challenging, such as different chaotic systems, chemical continuous stirred tank reactor, and power permanent magnet synchronous machine. Finally, software-in-the-loop (SiL) real-time simulations are presented to prove the practical application of the proposed approach.

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Šiljak, Dragoslav D., and Matija D. Šiljak. "Nonnegativity of uncertain polynomials." Mathematical Problems in Engineering 4, no. 2 (1998): 135–63. http://dx.doi.org/10.1155/s1024123x98000763.

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The purpose of this paper is to derive tests for robust nonnegativity of scalar and matrix polynomials, which are algebraic, recursive, and can be completed in finite number of steps. Polytopic families of polynomials are considered with various characterizations of parameter uncertainty including affine, multilinear, and polynomic structures. The zero exclusion condition for polynomial positivity is also proposed for general parameter dependencies. By reformulating the robust stability problem of complex polynomials as positivity of real polynomials, we obtain new sufficient conditions for robust stability involving multilinear structures, which can be tested using only real arithmetic. The obtained results are applied to robust matrix factorization, strict positive realness, and absolute stability of multivariable systems involving parameter dependent transfer function matrices.

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Ephremidze, L., I. Spitkovsky, and E. Lagvilava. "Rank-deficient spectral factorization and wavelets completion problem." International Journal of Wavelets, Multiresolution and Information Processing 13, no. 03 (May 2015): 1550013. http://dx.doi.org/10.1142/s0219691315500137.

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A simple constructive proof of polynomial matrix spectral factorization theorem is presented in the rank-deficient case. It is then used to provide an elementary solution to the wavelets completion problem.

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Csizmadia, Zsolt, Tibor Illés, and Adrienn Fico. "The s-monotone index selection rule for criss-cross algorithms of linear complementarity problems." Acta Universitatis Sapientiae, Informatica 5, no. 1 (July 1, 2013): 103–39. http://dx.doi.org/10.2478/ausi-2014-0007.

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Abstract In this paper we introduce the s-monotone index selection rules for the well-known criss-cross method for solving the linear complementarity problem (LCP). Most LCP solution methods require a priori information about the properties of the input matrix. One of the most general matrix properties often required for finiteness of the pivot algorithms (or polynomial complexity of interior point algorithms) is sufficiency. However, there is no known polynomial time method for checking the sufficiency of a matrix (classification of column sufficiency of a matrix is co-NP-complete). Following the ideas of Fukuda, Namiki and Tamura, using Existentially Polynomial (EP)-type theorems, a simple extension of the crisscross algorithm is introduced for LCPs with general matrices. Computational results obtained using the extended version of the criss-cross algorithm for bi-matrix games and for the Arrow-Debreu market equilibrium problem with different market size is presented.

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Grimble, M. J. "Polynomial matrix solution of the optimal deterministic continuous-time servomechanism problem." Transactions of the Institute of Measurement and Control 9, no. 4 (October 1987): 206–13. http://dx.doi.org/10.1177/014233128700900407.

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Kudin, Alexey, and Denis Vasilyev. "Counting real algebraic numbers with bounded derivative of minimal polynomial." International Journal of Number Theory 15, no. 10 (November 2019): 2223–39. http://dx.doi.org/10.1142/s1793042119501227.

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In this paper, we consider the problem of counting algebraic numbers [Formula: see text] of fixed degree [Formula: see text] and bounded height [Formula: see text] such that the derivative of the minimal polynomial [Formula: see text] of [Formula: see text] is bounded, [Formula: see text]. This problem has many applications to the problems of metric theory of Diophantine approximation. We prove that the number of [Formula: see text] defined above on the interval [Formula: see text] does not exceed [Formula: see text] for [Formula: see text] and [Formula: see text]. Our result is based on an improvement to a lemma from Gelfond’s monograph “Transcendental and algebraic numbers”. Given an integer polynomial small enough in some point, the lemma provides an upper bound for the absolute value of its irreducible divisor. We obtain a stronger estimate which holds in real points located far enough from all algebraic numbers of bounded degree and height. This is done by considering the resultant of two polynomials represented as the determinant of the Sylvester matrix for the shifted counterparts.

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Gervais, R., Q. I. Rahman, and G. Schmeisser. "(0, 2) – Interpolation of Entire Functions." Canadian Journal of Mathematics 38, no. 5 (October 1, 1986): 1210–27. http://dx.doi.org/10.4153/cjm-1986-061-2.

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Given a triangular matrix A whose nth row consists of the n points(1.1)Turán et al. ([12], [1], [2], [3]) considered the problem of existence, uniqueness, representation, convergence, etc. of polynomials f2n – 1 of degree ≧2n – 1 where the values of f2n – 1 and those of its second derivative are prescribed at the points (1.1), i.e.,(1.2)The choice of the points (1.1) is important. They found the zeros(1.3)of the polynomial(1.1)where Pn – 1 is the (n − 1) Legendre polynomial with the normalization Pn – 1(l) = 1 to be the most convenient.

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Brinkmeyer, Malte, Thasso Griebel, and Sebastian Böcker. "Polynomial Supertree Methods Revisited." Advances in Bioinformatics 2011 (December 21, 2011): 1–21. http://dx.doi.org/10.1155/2011/524182.

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Supertree methods allow to reconstruct large phylogenetic trees by combining smaller trees with overlapping leaf sets into one, more comprehensive supertree. The most commonly used supertree method, matrix representation with parsimony (MRP), produces accurate supertrees but is rather slow due to the underlying hard optimization problem. In this paper, we present an extensive simulation study comparing the performance of MRP and the polynomial supertree methods MinCut Supertree, Modified MinCut Supertree, Build-with-distances, PhySIC, PhySIC_IST, and super distance matrix. We consider both quality and resolution of the reconstructed supertrees. Our findings illustrate the tradeoff between accuracy and running time in supertree construction, as well as the pros and cons of voting- and veto-based supertree approaches. Based on our results, we make some general suggestions for supertree methods yet to come.

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Dhingra, A. K., A. N. Almadi, and D. Kohli. "A Gro¨bner-Sylvester Hybrid Method for Closed-Form Displacement Analysis of Mechanisms." Journal of Mechanical Design 122, no. 4 (August 1, 1999): 431–38. http://dx.doi.org/10.1115/1.1290395.

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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 that have been used to solve this problem. This paper presents a new approach to displacement analysis using the reduced Gro¨bner basis form of a system of equations under degree lexicographic (dlex) term ordering of its monomials and Sylvester’s Dialytic elimination method. Using the Gro¨bner-Sylvester hybrid approach, a finitely solvable system of equations F is transformed into its reduced Gro¨bner 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 condition for 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, 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 address the 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 I/O polynomial of an 8-link 1-DOF mechanism that does not contain any 4-link loop. [S1050-0472(00)01204-6]

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Behroozifar, Mahmoud, and Neda Habibi. "A numerical approach for solving a class of fractional optimal control problems via operational matrix Bernoulli polynomials." Journal of Vibration and Control 24, no. 12 (January 24, 2017): 2494–511. http://dx.doi.org/10.1177/1077546316688608.

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The purpose of this study is to introduce a novel approach based on the operational matrix of a Riemann–Liouville fractional integral of Bernoulli polynomials, in order to numerically solve a class of fractional optimal control problems that arise in engineering. The method is computationally consistent and moreover, it has good flexibility in satisfying the initial and boundary conditions. The fractional derivative in the dynamic system is considered in the Caputo sense. The upper bound of the error for function approximation by a Bernoulli polynomial is also given. In order to numerically solve the given problem, the problem is transformed into a functional integral equation that is equivalent to the given problem. Then, the new integral equation is approximated by utilizing the Gauss quadrature formula. Afterwards, a system of nonlinear equations is yielded from the Lagrange multipliers method. Finally, the resultant system of nonlinear equations is solved by Newton’s iterative method. Some illustrative examples are included to demonstrate the applicability of the new technique.

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Helmi, B. Hoda, Adel T. Rahmani, and Martin Pelikan. "A factor graph based genetic algorithm." International Journal of Applied Mathematics and Computer Science 24, no. 3 (September 1, 2014): 621–33. http://dx.doi.org/10.2478/amcs-2014-0045.

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Abstract We propose a new linkage learning genetic algorithm called the Factor Graph based Genetic Algorithm (FGGA). In the FGGA, a factor graph is used to encode the underlying dependencies between variables of the problem. In order to learn the factor graph from a population of potential solutions, a symmetric non-negative matrix factorization is employed to factorize the matrix of pair-wise dependencies. To show the performance of the FGGA, encouraging experimental results on different separable problems are provided as support for the mathematical analysis of the approach. The experiments show that FGGA is capable of learning linkages and solving the optimization problems in polynomial time with a polynomial number of evaluations.

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Bueno Cachadina, Maria Isabel, Javier Perez, Anthony Akshar, Daria Mileeva, and Remy Kassem. "Linearizations for Interpolatory Bases - a Comparison: New Families of Linearizations." Electronic Journal of Linear Algebra 36, no. 36 (December 18, 2020): 799–833. http://dx.doi.org/10.13001/ela.2020.5183.

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One strategy to solve a nonlinear eigenvalue problem $T(\lambda)x=0$ is to solve a polynomial eigenvalue problem (PEP) $P(\lambda)x=0$ that approximates the original problem through interpolation. Then, this PEP is usually solved by linearization. Because of the polynomial approximation techniques, in this context, $P(\lambda)$ is expressed in a non-monomial basis. The bases used with most frequency are the Chebyshev basis, the Newton basis and the Lagrange basis. Although, there exist already a number of linearizations available in the literature for matrix polynomials expressed in these bases, new families of linearizations are introduced because they present the following advantages: 1) they are easy to construct from the matrix coefficients of $P(\lambda)$ when this polynomial is expressed in any of those three bases; 2) their block-structure is given explicitly; 3) it is possible to provide equivalent formulations for all three bases which allows a natural framework for comparison. Also, recovery formulas of eigenvectors (when $P(\lambda)$ is regular) and recovery formulas of minimal bases and minimal indices (when $P(\lambda)$ is singular) are provided. The ultimate goal is to use these families to compare the numerical behavior of the linearizations associated to the same basis (to select the best one) and with the linearizations associated to the other two bases, to provide recommendations on what basis to use in each context. This comparison will appear in a subsequent paper.

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Cravo, Glória. "A Note on Matrix Completion Problems." Algebra Colloquium 19, spec01 (October 31, 2012): 1179–86. http://dx.doi.org/10.1142/s100538671200096x.

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Matrix completion problems are an important subclass of problems in matrix theory. An important question in matrix completion problems was posed by Oliveira in 1975, where the author proposed the description of the characteristic polynomial of a partitioned matrix of the form A = [Ai,j], i, j ∈ {1,2} (whose entries are in a field and A1,1, A2,2 are square submatrices), when some of the blocks Ai,j are prescribed and the others are unknown. The analysis of this problem gave rise to several subproblems, according to the location of the prescribed submatrices. Many authors have considered this list of subproblems. In this note we provide a new proof of a result obtained by Oliveira inserted in this class of subproblems.

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SUGIMOTO, KENJI, and YUTAKA YAMAMOTO. "New solution to the inverse regulator problem by the polynomial matrix method." International Journal of Control 45, no. 5 (May 1987): 1627–40. http://dx.doi.org/10.1080/00207178708933838.

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Grimble, M. J. "Polynomial matrix solution of H2 optimal control problem for state-space systems." Optimal Control Applications and Methods 23, no. 2 (2002): 59–89. http://dx.doi.org/10.1002/oca.703.

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Srivastava, Vineet. "NUMERICAL STUDY OF HEAT TRANSFERS IN TISSUES DURING HYPERTHERMIA USING MODIFIED BERNSTEIN POLYNOMIALS." Journal of Mathematical Sciences & Computational Mathematics 2, no. 1 (November 2, 2020): 129–44. http://dx.doi.org/10.15864/jmscm.2108.

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In the present article, we use modified Bernstein polynomial (B–polynomial) as a basis for the numerical approximation of heat transfer in hyperthermia treatment. A set of continuous polynomials over the spatial domain is use to expand the desired solution using discretization in time variable only. The Galerkin method is use to determine the expansion coefficients to construct initial trial functions. The system of equations has been solved using fourth–order Runge-Kutta method. The accuracy of the solutions is dependent on the size of the B–polynomial basis set. Also, Homotopy Perturbation Method has been applied to solve Matrix form of initial value differential equations which is transformed from boundary value differential equation of desired problem by using central difference scheme. The results thus obtained are in very good agreement with the previous results and it is presented graphically.

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35

Akyar, Handan, Taner Büyükköroğlu, and Vakıf Dzhafarov. "On Stability of Parametrized Families of Polynomials and Matrices." Abstract and Applied Analysis 2010 (2010): 1–16. http://dx.doi.org/10.1155/2010/687951.

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The Schur and Hurwitz stability problems for a parametric polynomial family as well as the Schur stability problem for a compact set of real matrix family are considered. It is established that the Schur stability of a family of real matrices is equivalent to the nonsingularity of the family if has at least one stable member. Based on the Bernstein expansion of a multivariable polynomial and extremal properties of a multilinear function, fast algorithms are suggested.

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Hernandez-Gonzalez, Miguel, and Michael V. Basin. "State estimation for stochastic polynomial systems with switching in the state equation." Transactions of the Institute of Measurement and Control 40, no. 9 (November 21, 2017): 2732–39. http://dx.doi.org/10.1177/0142331217737134.

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The problem of designing a mean-square filter has been studied for stochastic polynomial systems, where the state equation switches between two different nonlinear functions, over linear observations. A switching signal depends on a random variable modelled as a Bernoulli distributed sequence that takes the quantities of zero and one. The differential equations for the state estimate and the error covariance matrix are obtained in a closed form by expressing the conditional expectation of polynomial terms as functions of the estimate and covariance matrix. Finite-dimensional filtering equations are obtained for a particular case of a third-degree polynomial system. Numerical simulations are carried out in two cases of switching between different linear and second degree polynomial functions.

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Seo, Sang-hyup, Jong-Hyeon Seo, and Hyun-Min Kim. "Convergence of a modified Newton method for a matrix polynomial equation arising in stochastic problem." Electronic Journal of Linear Algebra 34 (February 21, 2018): 500–513. http://dx.doi.org/10.13001/1081-3810.3762.

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The Newton iteration is considered for a matrix polynomial equation which arises in stochastic problem. In this paper, it is shown that the elementwise minimal nonnegative solution of the matrix polynomial equation can be obtained using Newton's method if the equation satisfies the sufficient condition, and the convergence rate of the iteration is quadratic if the solution is simple. Moreover, it is shown that the convergence rate is at least linear if the solution is non-simple, but a modified Newton method whose iteration number is less than the pure Newton iteration number can be applied. Finally, numerical experiments are given to compare the effectiveness of the modified Newton method and the standard Newton method.

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Deng, Hongyao, Qingxin Zhu, Jinsong Tao, and Xiuli Song. "Optimizing Shrinkage Curves and Application in Image Denoising." Mathematical Problems in Engineering 2017 (2017): 1–13. http://dx.doi.org/10.1155/2017/4965262.

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A shrinkage curve optimization is proposed for weighted nuclear norm minimization and is adapted to image denoising. The proposed optimization method employs a penalty function utilizing the difference between a latent matrix and its observation and uses odd polynomials to shrink the singular values of the observation matrix. As a result, the coefficients of polynomial characterize the shrinkage operator fully. Furthermore, the Frobenius norm of the penalty function is converted into the corresponding spectral norm, and thus the parameter optimization problem can be easily solved by using off-and-shelf plain least-squares. In the practical application, the proposed denoising method does not work on the whole image at once, but rather a series of matrix termed Rank-Ordered Similar Matrix (ROSM). Simulation results on 256 noisy images demonstrate the effectiveness of the proposed algorithms.

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YUZBASI, SUAYIP, and NURCAN BAYKUS SAVASANERIL. "HERMITE POLYNOMIAL APPROACH FOR SOLVING SINGULAR PERTURBATED DELAY DIFFERENTIAL EQUATIONS." Journal of Science and Arts 20, no. 4 (December 30, 2020): 845–54. http://dx.doi.org/10.46939/j.sci.arts-20.4-a06.

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In this study, a collocation approach based on the Hermite polyomials is applied to solve the singularly perturbated delay differential eqautions by boundary conditions. By means of the matix relations of the Hermite polynomials and the derivatives of them, main problem is reduced to a matrix equation. And then, collocation points are placed in equation of the matrix. Hence, the singular perturbed problem is transformed into an algebraic system of linear equations. This system is solved and thus the coefficients of the assumed approximate solution are determined. Numerical applications are made for various values of N.

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Saeed, Umer, and Muhammad Umair. "A modified method for solving non-linear time and space fractional partial differential equations." Engineering Computations 36, no. 7 (August 12, 2019): 2162–78. http://dx.doi.org/10.1108/ec-01-2019-0011.

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Purpose The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain. Design/methodology/approach The proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations. Findings The fractional derivative of Lagrange polynomial is a big hurdle in classical DQM. To overcome this problem, the authors represent the Lagrange polynomial in terms of shifted Legendre polynomial. They construct a transformation matrix which transforms the Lagrange polynomial into shifted Legendre polynomial of arbitrary order. Then, they obtain the new weighting coefficients matrices for space fractional derivatives by shifted Legendre polynomials and use these in conversion of a non-linear fractional partial differential equation into a system of fractional ordinary differential equations. Convergence analysis for the proposed method is also discussed. Originality/value Many engineers can use the presented method for solving their time and space fractional non-linear partial differential equation models. To the best of the authors’ knowledge, the differential quadrature method has never been extended or implemented for non-linear time and space fractional partial differential equations.

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Han, Xiaobao, Zhenbao Liu, Huacong Li, and Xianwei Liu. "Output feedback controller design for polynomial linear parameter varying system via parameter-dependent Lyapunov functions." Advances in Mechanical Engineering 9, no. 2 (February 2017): 168781401769032. http://dx.doi.org/10.1177/1687814017690327.

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This article presents a new output feedback controller design method for polynomial linear parameter varying model with bounded parameter variation rate. Based on parameter-dependent Lyapunov function, the polynomial linear parameter varying system controller design is formulated into an optimization problem constrained by parameterized linear matrix inequalities. To solve this problem, first, this optimization problem is equivalently transformed into a new form with elimination of coupling relationship between parameter-dependent Lyapunov function, controller, and object coefficient matrices. Then, the control solving problem was reduced to a normal convex optimization problem with linear matrix inequalities constraint on a newly constructed convex polyhedron. Moreover, a parameter scheduling output feedback controller was achieved on the operating condition, which satisfies robust performance and dynamic performances. Finally, the feasibility and validity of the controller analysis and synthesis method are verified by the numerical simulation.

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Kim, D., and J. W. David. "An Improved Method for Stability and Damped Critical Speeds of Rotor-Bearing Systems." Journal of Vibration and Acoustics 112, no. 1 (January 1, 1990): 112–18. http://dx.doi.org/10.1115/1.2930086.

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Many computer programs are available for stability and critical speed analysis of rotor-bearing systems. The iterative search-transfer matrix method is widely used for such programs. However, this method sometimes fails to converge or may miss some critical speeds. The polynomial method, which derives the characteristic polynomial and solves for the critical speeds, can partly overcome these shortcomings. However, the advantage of the polynomial method disappears as the number of elements in the system increases. This is because the computational time required to find a characteristic polynomial increases exponentially with the number of elements. This paper describes an improved technique based on the transfer matrix-polynomial method, which reduces the computational time significantly and completely eliminates the possibility of missing some critical speeds. A new technique is developed for the derivation of the characteristic polynomial. The characteristic polynomial equation is then converted into an eigenvalue problem of its companion matrix, whose eigenvalues are identical with the roots of the polynomial equation. The process, which can find only some dominant eigenvalues, eliminates the possibility of missing some eigenvalues without any penalties in the computational time and accuracy. The results from the method presented here are compared with those from some other methods.

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Ghosal, Purnata, and B. V. Raghavendra Rao. "On Proving Parameterized Size Lower Bounds for Multilinear Algebraic Models." Fundamenta Informaticae 177, no. 1 (December 18, 2020): 69–93. http://dx.doi.org/10.3233/fi-2020-1980.

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We consider the problem of obtaining parameterized lower bounds for the size of arithmetic circuits computing polynomials with the degree of the polynomial as the parameter. We consider the following special classes of multilinear algebraic branching programs: 1) Read Once Oblivious Branching Programs (ROABPs), 2) Strict interval branching programs, 3) Sum of read once formulas with restricted ordering. We obtain parameterized lower bounds (i.e., nΩ(t(k)) lower bound for some function t of k) on the size of the above models computing a multilinear polynomial that can be computed by a depth four circuit of size g(k)nO(1) for some computable function g. Further, we obtain a parameterized separation between ROABPs and read-2 ABPs. This is obtained by constructing a degree k polynomial that can be computed by a read-2 ABP of small size such that the rank of the partial derivative matrix under any partition of the variables is large.

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Pugliese, Paolo. "On the normal matrix of the polynomial LS problem over the Chebyshev points." Linear Algebra and its Applications 378 (February 2004): 61–69. http://dx.doi.org/10.1016/j.laa.2003.09.002.

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45

Trzaska, Z. W. "The inverse problem to matrix polynomial factorisation and its application to circuit design." IEE Proceedings G (Electronic Circuits and Systems) 132, no. 5 (1985): 221. http://dx.doi.org/10.1049/ip-g-1.1985.0045.

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CAMPOS, RAFAEL G., and FRANCISCO DOMÍNGUEZ MOTA. "AN IMPLEMENTATION OF THE COLLOCATION METHOD FOR INITIAL VALUE PROBLEMS." International Journal of Modeling, Simulation, and Scientific Computing 04, no. 02 (June 2013): 1350006. http://dx.doi.org/10.1142/s1793962313500062.

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An implementation of the standard collocation method based on polynomial interpolation is presented in a matrix framework in this paper. The underlying differentiation matrix can be partitioned to yield a superconvergent implicit multistep-like method to solve the initial value problem numerically. The first- and second-order versions of this method are L-stable.

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47

Ottaviani, Giorgio. "An Invariant Regarding Waring’s Problem for Cubic Polynomials." Nagoya Mathematical Journal 193 (2009): 95–110. http://dx.doi.org/10.1017/s0027763000026040.

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AbstractWe compute the equation of the 7-secant variety to the Veronese variety (P4,O(3)), its degree is 15. This is the last missing invariant in the Alexander-Hirschowitz classification. It gives the condition to express a homogeneous cubic polynomial in 5 variables as the sum of 7 cubes (Waring problem). The interesting side in the construction is that it comes from the determinant of a matrix of order 45 with linear entries, which is a cube. The same technique allows to express the classical Aronhold invariant of plane cubics as a pfaffian.

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TAKADA, YUJI. "LEARNING EQUAL MATRIX GRAMMARS BASED ON CONTROL SETS." International Journal of Pattern Recognition and Artificial Intelligence 08, no. 02 (April 1994): 609–26. http://dx.doi.org/10.1142/s0218001494000322.

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An equal matrix grammar is a parallel rewriting system. In this paper, we consider the problem of learning equal matrix grammars from examples. We introduce a learning method based on control sets and show two subclasses learnable in polynomial time with learning methods for regular sets. We also show that for any equal matrix language there exists an equal matrix grammar learnable efficiently from positive structural examples only.

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Matos, José M. A., and Maria João Rodrigues. "Almost Exact Computation of Eigenvalues in Approximate Differential Problems." Mathematical and Computational Applications 24, no. 4 (November 14, 2019): 96. http://dx.doi.org/10.3390/mca24040096.

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Differential eigenvalue problems arise in many fields of Mathematics and Physics, often arriving, as auxiliary problems, when solving partial differential equations. In this work, we present a method for eigenvalues computation following the Tau method philosophy and using Tau Toolbox tools. This Matlab toolbox was recently presented and here we explore its potential use and suitability for this problem. The first step is to translate the eigenvalue differential problem into an algebraic approximated eigenvalues problem. In a second step, making use of symbolic computations, we arrive at the exact polynomial expression of the determinant of the algebraic problem matrix, allowing us to get high accuracy approximations of differential eigenvalues.

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WERNICKE, SEBASTIAN, JOCHEN ALBER, JENS GRAMM, JIONG GUO, and ROLF NIEDERMEIER. "THE COMPUTATIONAL COMPLEXITY OF AVOIDING FORBIDDEN SUBMATRICES BY ROW DELETIONS." International Journal of Foundations of Computer Science 17, no. 06 (December 2006): 1467–84. http://dx.doi.org/10.1142/s0129054106004522.

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We initiate a systematic study of the ROW DELETION(B) problem on matrices: Given an input matrix A and a fixed "forbidden submatrix" B, the task is to remove a minimum number of rows from A such that no row or column permutation of B occurs as a submatrix in the resulting matrix. An application of this problem can be found, for instance, in the construction of perfect phylogenies. Establishing a strong connection to variants of the NP-complete HITTING SET problem, we describe and analyze structural properties of B that make ROW DELETION(B)NP-complete. On the positive side, the close relation with HITTING SET problems yields constant-factor polynomial-time approximation algorithms and fixed-parameter tractability results.

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Journal articles on the topic 'Matrix Polynomial Problem'

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