Showing papers on "Matrix differential equation published in 2014"
TL;DR: In this paper, the modified Korteweg-de-Vries equation in terms of Wronskians is reviewed and the obtained solutions are categorized into two types: solitons and breathers, together with their limit cases.
Abstract: This is a continuation of [Notes on solutions in Wronskian form to soliton equations: Korteweg–de Vries-type, arXiv:nlin.SI/0603008]. In the present paper, we review solutions to the modified Korteweg–de Vries equation in terms of Wronskians. The Wronskian entry vector needs to satisfy a matrix differential equation set which contains complex operation. This fact makes the analysis of the modified Korteweg–de Vries to be different from the case of the Korteweg–de Vries equation. To derive complete solution expressions for the matrix differential equation set, we introduce an auxiliary matrix to deal with the complex operation. As a result, the obtained solutions to the modified Korteweg–de Vries equation are categorized into two types: solitons and breathers, together with their limit cases. Besides, we give rational solutions to the modified Korteweg–de Vries equation in Wronskian form. This is derived with the help of a Galilean transformed version of the modified Korteweg–de Vries equation. Finally, typical dynamics of the obtained solutions are analyzed and illustrated. We also list out the obtained solutions and their corresponding basic Wronskian vectors in the conclusion part.
85 citations
TL;DR: Several new results are established to express the structures and bounds of the eigenvalues related to the symmetric positive definite matrices and a family of iterative algorithms are presented for the matrix equation AX=F and the coupled Sylvester matrix equations.
Abstract: In this paper, we discuss the properties of the eigenvalues related to the symmetric positive definite matrices Several new results are established to express the structures and bounds of the eigenvalues Using these results, a family of iterative algorithms are presented for the matrix equation AX=F and the coupled Sylvester matrix equations The analysis shows that the iterative solutions given by the least squares based iterative algorithms converge to their true values for any initial conditions The effectiveness of the proposed iterative algorithm is illustrated by a numerical example
75 citations
TL;DR: The global convergence of the algorithm is proved and it is shown that it can be effectively used for the minimization of extreme eigenvalues, e.g., the largest eigenvalue or the sum of the largest specified number of eigen values.
Abstract: This work concerns the global minimization of a prescribed eigenvalue or a weighted sum of prescribed eigenvalues of a Hermitian matrix-valued function depending on its parameters analytically in a box. We describe how the analytical properties of eigenvalue functions can be put into use to derive piecewise quadratic functions that underestimate the eigenvalue functions. These piecewise quadratic underestimators lead us to a global minimization algorithm, originally due to Breiman and Cutler. We prove the global convergence of the algorithm and show that it can be effectively used for the minimization of extreme eigenvalues, e.g., the largest eigenvalue or the sum of the largest specified number of eigenvalues. This is particularly facilitated by the analytical formulas for the first derivatives of eigenvalues, as well as analytical lower bounds on the second derivatives that can be deduced for extreme eigenvalue functions. The applications that we have in mind also include the ${\rm H}_\infty$-norm of a ...
60 citations
TL;DR: A procedure for approximating fractional-order systems by means of integer-order state-space models is presented, based on the rational approximation of fractiona-order operators suggested by Oustaloup.
Abstract: A procedure for approximating fractional-order systems by means of integer-order state-space models is presented. It is based on the rational approximation of fractional-order operators suggested by Oustaloup. First, a matrix differential equation is obtained from the original fractional-order representation. Then, this equation is realized in a state-space form that has a sparse block-companion structure. The dimension of the resulting integer-order model can be reduced using an efficient algorithm for rational L2 approximation. Two numerical examples are worked out to show the performance of the suggested technique.
54 citations
TL;DR: In this paper, the authors derived new bounds on the extreme eigenvalues of a spatial correlation matrix that is characterized by the exponential model in a massive MIMO system and showed that these bounds can be exploited to analyze many wireless communication scenarios including uniform planar arrays.
Abstract: It is critical to understand the properties of spatial correlation matrices in massive multiple-input-multiple-output (MIMO) systems. We derive new bounds on the extreme eigenvalues of a spatial correlation matrix that is characterized by the exponential model in this paper. The new upper bound on the maximum eigenvalue is tighter than the previously known bound. Moreover, numerical studies show that our new lower bound on the maximum eigenvalue is close to the true maximum eigenvalue in most cases. We also derive an upper bound on the minimum eigenvalue that is also tight. These bounds can be exploited to analyze many wireless communication scenarios including uniform planar arrays, which are expected to be widely used for massive MIMO systems.
49 citations
TL;DR: The relation of the two approaches to dynamically orthogonal field equations is examined and it is proved theoretically and numerically their equivalence, in the sense that one method is an exact reformulation of the other.
38 citations
TL;DR: In this paper, a nonlinear eigenvalue problem for an autonomous ordinary differential equation of the second order is considered, and a new method that allows one to derive an equation w.r.t. spectral parameter (the dispersion equation) which contains all necessary information about the eigenvalues is presented.
Abstract: In this work a nonlinear eigenvalue problem for a nonlinear autonomous ordinary differential equation of the second order is considered. This problem describes the process of propagation of transverse-electric electromagnetic waves along a plane dielectric waveguide with nonlinear permittivity. We demonstrate, as far as we know, a new method that allows one to derive an equation w.r.t. spectral parameter (the dispersion equation) which contains all necessary information about the eigenvalues. The method is based on a simple idea that the distance between zeros of a periodic solution to the differential equation is the same for the adjacent zeros. This method has no connections with the perturbation theory or the notion of a bifurcation point. Theorem of equivalence between the eigenvalue problem and the dispersion equation is proved. Periodicity of the eigenfunctions is proved, a formula for the period is found, and zeros of the eigenfunctions are determined. The formula for the distance between adjacent zeros of any eigenfunction is given. Also theorems of existence and localization of the eigenvalues are proved.
32 citations
TL;DR: The structure of the spectral solutions of the nonlinear matrix equation A X A = X A X by showing that any semisimple eigenvalue of A with multiplicity at least 2 gives rise to infinitely many solutions.
29 citations
TL;DR: In this paper, the eigenvalues of an unknown density matrix of a finite-dimensional system in a single experimental setting are determined with the minimal number of parameters obtained by a measurement of a single observable.
Abstract: Eigenvalues of a density matrix characterize well the quantum state's properties, such as coherence and entanglement. We propose a simple method to determine all the eigenvalues of an unknown density matrix of a finite-dimensional system in a single experimental setting. Without fully reconstructing a quantum state, eigenvalues are determined with the minimal number of parameters obtained by a measurement of a single observable. Moreover, its implementation is illustrated in linear optical and superconducting systems.
23 citations
TL;DR: The practical features of the proposed methods are that it is implemented in the second-order setting itself using only those small number of eigenvalues and the eigenvectors that are to be assigned and the no spill-over is established by means of mathematical results.
23 citations
TL;DR: In this article, a new approach based on the Adomian decomposition method and the Faddeev-Leverrier algorithm is presented for finding real eigenvalues of any desired real matrices.
TL;DR: A similarity transformation matrix is introduced which better reveals the eigenvalues of the admittance matrix and is shown to accurately capture the large and small eigen values alike, thereby avoiding error magnifications in time-domain simulations.
Abstract: Modeling of frequency-dependent components and subnetworks is often based on a terminal description by an admittance matrix in the frequency domain. One challenge in the extraction of state-space models from such data is to prevent possible error magnification when the model is to be applied in time-domain simulations. The error magnification is a consequence of inaccurate representation of small eigenvalues of the admittance matrix. This paper resolves the problem by introducing a similarity transformation matrix which better reveals the eigenvalues of the admittance matrix. The chosen transformation preserves the passivity and symmetry of the original data, allowing the modeling to be performed by standard methods for model extraction and passivity enforcement. The approach is demonstrated for the wideband modeling of subnetworks and power transformers. The new technique is shown to accurately capture the large and small eigenvalues alike, thereby avoiding error magnifications in time-domain simulations.
06 Apr 2014
TL;DR: To mitigate this problem, this work has developed a tracking algorithm to keep the ordering of modes as consistent as possible over frequency.
Abstract: Characteristic mode analysis (CMA) is a useful design tool that enables antenna designers to follow a nonbrute force approach of systematically extracting the radiating properties of a structure. These properties are quantified with CMA in the form of eigenvalues and eigenvectors, i.e., the solutions of a generalised eigenvalue equation that is formulated from the Method-of-Moments (MoM) impedance matrix. One particular challenge to CMA, however, is the manner in which quantities such as eigenvalues are plotted as a function of frequency. At each discrete frequency sample, the eigenvalues are sorted according to modal significance, i.e., based on the efficiency with which they radiate. At higher frequencies, the ordering of modes may differ from that obtained at lower frequencies. To mitigate this problem, we have developed a tracking algorithm to keep the ordering of modes as consistent as possible over frequency.
TL;DR: It is shown that, for the spikes located above a phase transition threshold, the asymptotic behavior of the log ratio of the joint density of the eigenvalues of the F matrix to their joint density under a local deviation from these values depends only on the k of the largest eigen values of the corresponding F matrix.
Abstract: We consider two types of spiked multivariate F distributions: a scaled distribution with the scale matrix equal to a rank-one perturbation of the identity, and a distribution with trivial scale, but rank-one non-centrality. The norm of the rank-one matrix (spike) parameterizes the joint distribution of the eigenvalues of the corresponding F matrix. We show that, for a spike located above a phase transition threshold, the asymptotic behavior of the log ratio of the joint density of the eigenvalues of the F matrix to their joint density under a local deviation from this value depends only on the largest eigenvalue λ1. Furthermore, λ1 is asymptotically normal, and the statistical experiment of observing all the eigenvalues of the F matrix converges in the Le Cam sense to a Gaussian shift experiment that depends on the asymptotic mean and variance of λ1. In particular, the best statistical inference about a sufficiently large spike in the local asymptotic regime is based on the largest eigenvalue only. As a by-product of our analysis, we establish joint asymptotic normality of a few of the largest eigenvalues of the multi-spiked F matrix when the corresponding spikes are above the phase transition threshold.
TL;DR: In this article, the existence theorem for the exterior Dirichlet problems for a class of fully nonlinear elliptic equations, which are related to the eigenvalues of the Hessian matrix, with prescribed asymptotic behavior at infinity was established.
TL;DR: For all 1 ≤ k ≤ n, | λ 1 ⋯ λ k | ≤ C n, k γ 1 ⊯ γ k, where C n, k is a combinatorial constant depending only on k and on the pattern of the matrix as mentioned in this paper.
TL;DR: An exact analytical expression is presented for all the eigenvalues of the Markov matrix of a class of scale-free polymer networks and it is used to derive an explicit formula for the random target access time for random walks on the studied networks.
Abstract: Much important information about the structural and dynamical properties of complex systems can be extracted from the eigenvalues and eigenvectors of a Markov matrix associated with random walks performed on these systems, and spectral methods have become an indispensable tool in the complex system analysis. In this paper, we study the Markov matrix of a class of scale-free polymer networks. We present an exact analytical expression for all the eigenvalues and determine explicitly their multiplicities. We then use the obtained eigenvalues to derive an explicit formula for the random target access time for random walks on the studied networks. Furthermore, based on the link between the eigenvalues of the Markov matrix and the number of spanning trees, we confirm the validity of the obtained eigenvalues and their corresponding degeneracies.
TL;DR: It is proved that the discrete solution resulting from the linear system converges exponentially to the true solution of the order‐reduced system of ODEs, and the eigenvalues of certain approximation to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the discretized linear system.
Abstract: By introducing a variable substitution we transform the two-point boundary value problem of a third-order ordinary differential equation into a system of two second-order ordinary differential equations. We discretize this order-reduced system of ordinary differential equations by both sinc-collocation and sinc-Galerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the order-reduced system of ordinary differential equations. The coefficient matrix of the linear system is of block two-by-two structure and each of its blocks is a combination of Toeplitz
TL;DR: In this article, an analytical representation for the solution of the neutron point kinetics equation, free of stiffness and assuming that reactivity is a continuous or sectionally continuous function of time, is presented.
TL;DR: In this paper, a differential equation for K Bessel function was solved and the relationship between Bessel functions and K -Bessel functions was established. And the generating function for KBessel functions were evaluated.
Abstract: In this paper we solve a differential equation for K Bessel function. We establish a relationship between Bessel function and K -Bessel function. Finally we evaluate the generating function for KBessel function. Mathematics Subject Classification: 33B15, 33C20, 34B30
TL;DR: Two inverse eigenvalue problems are solved: given spectral properties, the two basis Krylov matrices linked to the nonsymmetric Lanczos algorithm are reconstructed and the extended tridiagonal matrix captures the recurrence coefficients of bi-orthogonal rational functions.
TL;DR: A FORTRAN 77 program is presented for calculating with the given accuracy eigenvalues, surface eigenfunctions and their first derivatives with respect to a parameter of the parametric self-adjoined 2D elliptic partial differential equation with the Dirichlet and/or Neumann type boundary conditions on a finite two-dimensional region.
TL;DR: In this article, an improved globally exponential stability criterion of a certain neutral delayed differential equation with time-varying of the form has been proposed in the form of linear matrix inequality.
Abstract: In this paper, an improved globally exponential stability criterion of a certain neutral delayed differential equation with time-varying of the form has been proposed in the form of linear matrix inequality. We first propose an upper bound of the solution in terms of an exponential function. Then we apply Lyapunov functions, a descriptor form, the Leibniz-Newton formula and radially unboundedness to formulate the sufficient criterion. To show the effectiveness of the proposed criterion, four numerical examples are presented.
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TL;DR: In this article, the solvability of the Sylvester-like matrix equation through an auxiliary standard (or generalized) generalized sylvester matrix equation is discussed and the closed-form solutions can be found by using previous results.
Abstract: Many applications in applied mathematics and control theory give rise to the unique solution of a Sylvester-like matrix equation associated with an underlying structured matrix operator $f$. In this paper, we will discuss the solvability of the Sylvester-like matrix equation through an auxiliary standard (or generalized) Sylvester matrix equation. We also show that when this Sylvester-like matrix equation is uniquely solvable, the closed-form solutions can be found by using previous result. In addition, with the aid of the Kronecker product some useful results of the solvability of this matrix equation are provided.
TL;DR: In this article, the authors studied the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u∂u/∂x.
Abstract: We study the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u∂u/∂x. Our aim is to find the most general nontrivial form of the dispersion relation ω(k) for which the five-wave interaction scattering matrix is identically zero on the resonance manifold. As could be expected, the matrix in one dimension is zero for the Korteweg-de Vries equation, the Benjamin-Ono equation, and the intermediate long-wave equation. In two dimensions, we find a new equation that satisfies our requirement.
TL;DR: In this article, the spectral problem of a class of fractional differential equations from nonlocal continuum mechanics was studied and the spectral spectrum of this problem was shown to have only countable real eigenvalues with finite multiplicity.
Abstract: This paper studies the spectral problem of a class of fractional differential equations from nonlocal continuum mechanics. By applying the spectral theory of compact self-adjoint operators in Hilbert spaces, we show that the spectrum of this problem consists of only countable real eigenvalues with finite multiplicity and the corresponding eigenfunctions form a complete orthogonal system. Furthermore, we obtain the lower bound of the eigenvalues.
01 Jan 2014
TL;DR: In this paper, a matrix representation for the differential resultant of two generic ordinary differential polynomials in the differential indeterminate (y) with order one and arbitrary degree is given.
Abstract: In this paper, a matrix representation for the differential resultant of two generic ordinary differential polynomials \(f_1\) and \(f_2\) in the differential indeterminate \(y\) with order one and arbitrary degree is given. That is, a nonsingular matrix is constructed such that its determinant contains the differential resultant as a factor. Furthermore, the algebraic sparse resultant of \(f_1, f_2, {\updelta } f_1\) and \({\updelta } f_2\) treated as polynomials in \(y, y^{\prime }, y^{\prime \prime }\) is shown to be a nonzero multiple of the differential resultant of \(f_1\) and \(f_2\). Although very special, this seems to be the first matrix representation for a class of nonlinear generic differential polynomials.
TL;DR: In the spirit of the Rayleigh conjecture for the biharmonic operator, it is proved that balls are critical points with volume constraint for all simple eigenvalues and the elementary symmetric functions of multiple eigen values.
Abstract: We consider the eigenvalue problem for the Reissner-Mindlin system arising in the study of the free vibration modes of an elastic clamped plate. We provide quantitative estimates for the variation of the eigenvalues upon variation of the shape of the plate. We also prove analyticity results and establish Hadamard-type formulas. Finally, we address the problem of minimization of the eigenvalues in the case of isovolumetric domain perturbations. In the spirit of the Rayleigh conjecture for the biharmonic operator, we prove that balls are critical points with volume constraint for all simple eigenvalues and the elementary symmetric functions of multiple eigenvalues.
TL;DR: In this paper, the first-order perturbation of structural vibration eigenvalues and eigenvectors is derived on the basis of the matrix-perturbation theory when structural parameters such as stiffness and mass have changed.
Abstract: It has been extensively recognized that the engineering structures are becoming increasingly precise and complex, which makes the requirements of design and analysis more and more rigorous. Therefore the uncertainty effects are indispensable during the process of product development. Besides, iterative calculations, which are usually unaffordable in calculative efforts, are unavoidable if we want to achieve the best design. Taking uncertainty effects into consideration, matrix perturbation methodpermits quick sensitivity analysis and structural dynamic re-analysis, it can also overcome the difficulties in computational costs. Owing to the situations above, matrix perturbation method has been investigated by researchers worldwide recently. However, in the existing matrix perturbation methods, correlation coefficient matrix of random structural parameters, which is barely achievable in engineering practice, has to be given or to be assumed during the computational process. This has become the bottleneck of application for matrix perturbation method. In this paper, we aim to develop an executable approach, which contributes to the application of matrix perturbation method. In the present research, the first-order perturbation of structural vibration eigenvalues and eigenvectors is derived on the basis of the matrix perturbation theory when structural parameters such as stiffness and mass have changed. Combining the first-order perturbation of structural vibration eigenvalues and eigenvectors with the probability theory, the variance of structural random eigenvalue is derived from the perturbation of stiffness matrix, the perturbation of mass matrix and the eigenvector of baseline-structure directly. Hence the Direct-Variance-Analysis (DVA) method is developed to assess the variation range of the structural random eigenvalues without correlation coefficient matrix being involved. The feasibility of the DVA method is verified with two numerical examples (one is truss-system and the other is wing structure of MA700 commercial aircraft), in which the DVA method also shows superiority in computational efficiency when compared to the Monte-Carlo method.
TL;DR: In this article, a new approach is presented for obtaining the solutions to Yakubovich-conjugate quaternion matrix equation based on the real representation of quaternions.
Abstract: A new approach is presented for obtaining the solutions to Yakubovich--conjugate quaternion matrix equation based on the real representation of a quaternion matrix. Compared to the existing results, there are no requirements on the coefficient matrix . The closed form solution is established and the equivalent form of solution is given for this Yakubovich--conjugate quaternion matrix equation. Moreover, the existence of solution to complex conjugate matrix equation is also characterized and the solution is derived in an explicit form by means of real representation of a complex matrix. Actually, Yakubovich-conjugate matrix equation over complex field is a special case of Yakubovich--conjugate quaternion matrix equation . Numerical example shows the effectiveness of the proposed results.