Showing papers on "Matrix differential equation published in 1994"

How many eigenvalues of a random matrix are real

Abstract: Let A be an n x n matrix whose elements are independent random variables with standard normal distributions. As n oo , the expected number of real eigenvalues is asymptotic to V/7r . We obtain a closed form expres- sion for the expected number of real eigenvalues for finite n , and a formula for the density of a real eigenvalue for finite n . Asymptotically, a real normalized eigenvalue Al/ii of such a random matrix is uniformly distributed on the in- terval [1, 1] . Analogous, but strikingly different, results are presented for the real generalized eigenvalues. We report on numerical experiments confirming these results and suggesting that the assumption of normality is not important for the asymptotic results. DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MASSACHUSETTS 02139 E-mail address: edelmanfmath . mit . edu ARTS AND SCIENCES, KAPI'OLANI COMMUNITY COLLEGE, 4303 DIAMOND HEAD ROAD, HONOLULU, HAWAII 98616 E-mail address: kostlanfkahuna .math. hawaii. edu TJ WATSON RSEARCH CENTER, 32-2, IBM, YORKTOWN HEIGHTS, NEW YORK 10598-0218 E-mail address: shubQwatson. ibm. com This content downloaded from 157.55.39.29 on Tue, 12 Apr 2016 08:54:01 UTC All use subject to http://about.jstor.org/terms

Exact solution for the distribution of transmission eigenvalues in a disordered wire and comparison with random-matrix theory.

Abstract: An exact solution is presented of the Fokker-Planck equation which governs the evolution of an ensemble of disordered metal wires of increasing length, in a magnetic field. By a mapping onto a free-fermion problem, the complete probability distribution function of the transmission eigenvalues is obtained. The logarithmic eigenvalue repulsion of random-matrix theory is shown to break down for transmission eigenvalues which are not close to unity. ***Submitted to Physical Review B.****

Numerical methods for eighth-, tenth- and twelfth-order eigenvalue problems arising in thermal instability

Abstract: Second-order finite-difference methods are developed for the numerical solutions of the eighth-, tenth- and twelfth-order eigenvalue problems arising in the study of the effect of rotation on a horizontal layer of fluid heated from below. Instability setting-in as overstability may be modelled by an eighth-order ordinary differential equation. When a uniform magnetic field also acts across the fluid in the same direction as gravity, instability setting-in as ordinary convection may be modelled by a tenth-order differential equation, while instability setting-in as overstability may be modelled by a twelfth-order differential equation. The numerical methods are developed by making direct replacements of the derivatives in the differential equations and then by computing the eigenvalues, which may incorporate Rayleigh number, horizontal wave speed and a time constant, from the resulting algebraic eigenvalue problem. The eigenvalues are also computed by writing the differential equations as systems of second-order differential equations and then using second- and fourth-order methods to obtain the eigenvalues. Numerical results obtained using the two approaches are compared with estimates appearing in the literature.

Exact nth Derivatives of Eigenvalues and Eigenvectors

Abstract: Computing derivatives of eigenvalues and eigenvectors is of considerable importance in mathematics, physics, and engineering. These derivatives are essential for sensitivity analysis, which is used to study the effect of change in various parameters of an eigensystem on its performance. General solutions of eigenfunction s' derivatives have not been available. In this paper, exact analytical solutions for the wth derivatives of unrepeated eigenvalues and corresponding eigenvectors are given for general nonlinear and linear eigenvalue problems. The application of the theory is illustrated with examples.

Linear stochastic differential equations and Wick products

Abstract: We establish the existence and uniqueness of the solution to a multidimensional linear Skorohod stochastic differential equation with deterministic diffusion matrix, using the notions of Wick product andStransform. If the diffusion matrix is constant and has real eigenvalues, the solution is a stochastic process with moments of all orders, provided that the initial condition is differentiable up to a suitable order. The case of a diffusion matrix in the first Wiener chaos is discussed in the last section.

Analytic solutions for the finite‐difference time‐domain and transmission‐line‐matrix methods

Abstract: Eigenmodal decomposition formulations are given for numerical solutions of the finite-difference time-domain (FDTD) and the transmission-line-matrix (TLM) methods. Instead of direct simulation with these time-recursive schemes, the analysis involves two steps: (1) solving an eigenvalue problem, and (2) analytically constructing the numerical solutions in terms of the eigenvalues and eigenvectors. The numerical solution at any time step can be obtained with only O(N) computation once the corresponding eigenvalue problem has been solved. The main advantage of this technique is that the eigenvalues and eigenvectors for a problem can be stored, the numerical solutions then quickly processed with the stored data. In addition, high-frequency numerical noise can be reduced simply by discarding the related high-frequency modes. © 1994 John Wiley & Sons, Inc.

Eigenvalues of convex processes and convergence properties of differential inclusions

Abstract: In this paper, we study (real) eigenvalues and eigenvectors of convex processes, and provide conditions for the existence of eigenvectors in a given convex coneK⊂ℝn. It is established that the maximal eigenvalue ofG(·) inK is expressed by (whereK0 is the polar cone ofK) provided that the minimum is attained in intK0. This result is applied to study the asymptotic behaviour of certain differential inclusions{∈G(x(t)). We extend some known results for the von Neumann-Gale model to our more general framework. We prove that ifx0 is the unique eigenvector corresponding to the maximal eigenvalue λ0 ofG(·) inK, then the nonexistence of solutions of a certain special trigonometric form is necessary and sufficient for every viable solutionx(·) to satisfy-λ0tx(t)→cx0 ast←∞ for somec≥0. Our method is to study the family of convex conesWλ=cl{v−λx :x∈K,v∈G(x) where λ is any real number. We characterize the maximal eigenvalue λ0 as the minimal λ for whichWλ can be separated fromK.

Iterative computation of second‐order derivatives of eigenvalues and eigenvectors

Abstract: An iterative method is introduced for computing second-order partial derivatives (sensitivities) of eigenvalues and eigenvectors of matrices which depend on a number of real design parameters. Numerical tests confirm the viability of the method and support our theoretical analysis. Alternative methods are reviewed briefly and compared with the one proposed here.

Approximate eigenvalues, eigenvectors and inverse of a matrix with polynomial entries

Abstract: LetM be a matrix with polynomial entries This paper deals with calculating eigenvalues, eigenvectors and inverse ofM approximately in the form of truncated power series The calculation of approximate eigenvalues is based on algorithms solving algebraic equation symbolically in terms of power series Then, we give algorithms for calculating eigenvectors and inverse in terms of power series approximately We show an example for each algorithm Finally, we explain acceleration of convergence of power series and estimate of errors due to truncation of power series

Use of eigenvectors in understanding and correcting storage ring orbits

Abstract: The response matrix A is defined by the equation X = AΘ, where Θ is the kick vector and X is the resulting orbit vector. Since A is not necessarily a symmetric or even a square matrix we symmetrize it by using ATA. Then we find the eigenvalues and eigenvectors of this ATA matrix. The physical interpretation of the eigenvectors for circular machines is discussed. The task of the orbit correction is to find the kick vector Θ for a given measured orbit vector X. We are presenting a method, in which the kick vector is expressed as linear combination of the eigenvectors. An additional advantage of this method is that it yields the smallest possible kick vector to correct the orbit. We will illustrate the application of the method to the NSLS X-ray and UV storage rings and the resulting measurements. It will be evident, that the accuracy of this method allows the combination of the global orbit correction and local optimization of the orbit for beam lines and insertion devices. The eigenvector decomposition can also be used for optimizing kick vectors, taking advantage of the fact that eigenvectors with corresponding small eigenvalues generate negligible orbit changes. Thus, one can reduce a kick vector calculated by any other correction method and still stay within the tolerance for orbit correction. The use of eigenvectors in accurately measuring the response matrixand the use of the eigenvalue decomposition orbit correction algorithm in digital feedback is discussed.

Bounds for departure from normality and the Frobenius norm of matrix eigenvalues

Abstract: New lower and upper bounds for the departure from normality and the Frobenius norm of the eigenvalues of a matrix axe given. The significant properties of these bounds axe also described. For example, the upper bound for matrix eigenvalues improves upon the one derived by Kress, de Vries and Wegmann in [Lin. Alg. Appl., 8 (1974), pp. 109-120]. The upper bound for departure from normality is sharp for any matrix whose eigenvalues are collinear in the complex plane. Moreover, the latter formula is a practical estimate that costs (at most) 2m multiplications, where m is the number of nonzeros in the matrix. In terms of applications, the results can be used to bound from above the sensitivity of eigenvalues to matrix perturbations or bound from below the condition number of the eigenbasis of a matrix.

A simple test to determine the stability of an interval matrix

Abstract: Describes a sufficient condition by means of a simpler test that guarantees stability of a linear time-invariant system with parametric uncertainty in the "A" matrix. The parametric uncertainty is represented by an interval matrix. The proposed test is simpler than the existing ones. It is based on the eigenvalues of the hermitian part of L and P=(U-L), an upper bound /spl xi/, is derived from their maximum eigenvalues. The result presented is for the general case of an interval matrix. Simple examples are shown to illustrate its practical application.

Regulator theory for switched systems

Abstract: The exact output tracking problem for systems with parameters' jumps is considered. Necessary and sufficient conditions are derived for the elimination of switching-introduced output transient. Previous works have studied this problem for minimum-phase systems. In contrast, our approach, which is also applicable for nonminimum-phase systems, obtains bounded but possibly noncausal solutions. A major advantage, for the case of reference trajectory generated by a "switched" linear exosystem, is that a pure feedback solution is obtained, A linear map form the states of the exosystem to the desired system state is defined via a matrix differential equation whose constant solution provides asymptotic tracking, and coincides with the feedback law used in standard regulator theory. The obtained results are applied to a simple flexible structure with jumps in the pay-load mass. >

Nondiagonable systems under small perturbations

Abstract: By a nondiagonable system, we mean a system whose state matrix is nondiagonable, i.e. having nonlinear elementary divisors. In this paper, closed form formulae are given for the shifts occuring in the eigenvalues and eigenvectors of these types of systems due to small variations in the system elements.

Negative Eigenvalue Counting Method for One-Dimensional Mesoscopic Rings

Abstract: Negative eigenvalue counting method has been extended to deal with the closed ring with large number of atoms threaded by a magnetic flux This new method is extremely computer memory-saving and can be used to accurately determine the eigenvalues and eigenvectors As numerical examples, the results of second moment and persistent current are presented

Matrix method for eigenvalue assignment: The single input case

Abstract: The eigenvalue assignment/pole placement procedure has found application in a wide variety of control problems. The associated literature is rather extensive with a number of techniques discussed to that end. In this paper a method for assigning eigenvalues to a Linear Time Invariant (LTI) single input system is proposed. The algorithm determines a matrix, which has eigenvalues at the desired locations. It is obtained from the knowledge of the open-loop system and the desired eigenvalues. Solution of the matrix equation, involving unknown controller gains, open-loop system matrices and desired eigenvalues, results in the state feedback controller. The proposed algorithm requires the closed-loop eigenvalues to be different from those of the open-loop case. This apparent constraint is easily overcome by a negligible shift in the values. Two examples are considered to verify the proposed algorithm. The first one pertains to the in-plane libration of a Tethered Satellite System (TSS) while the second is concerned with control of the short period dynamics of a flexible airplane. Finally, the method is extended to determine the Controllability Grammian, corresponding to the specified closed-loop eigenvalues, without computing the controller gains.

Numerical treatment of the first-order pll equation with sinusoidal phase detector characteristic

Abstract: We discuss a powerful method depending on complex continued fractions to obtain and to solve a nonlinear equation for all eigenvalues of the underlying boundary value problem of the first-order phase locked loop equation. Furthermore we give several numerical examples and - for the sake of comparison - we list the first unverified and verified eigenvalues for a relevant signal-to-noise ratio.