Showing papers on "Matrix differential equation published in 1995"

A $q$-anaolg of the sixth Painlev\'e equation

Abstract: A $q$-difference analog of the sixth Painleve equation is presented. It arises as the condition for preserving the connection matrix of linear $q$-difference equations, in close analogy with the monodromy preserving deformation of linear differential equations. The continuous limit and special solutions in terms of $q$-hypergeometric functions are also discussed.

Examination, clarification, and simplification of modal decoupling method for multiconductor transmission lines

Abstract: In the application of the modal decoupling method, questions arise as to why the nonnormal matrices LC and CL are diagonalizable. Is the definition of the characteristic impedance matrix Z/sub c/ unique? Is it possible to normalize current and voltage eigenvectors simultaneously, yet assure the correct construction of the Z/sub c/ matrix? Under what conditions do M/sub i//sup t/M/sub u/=I and Z/sub c/=M/sub u/M/sub i//sup -1/? In this paper, these questions are thoroughly addressed. We prove the diagonalizability of matrices LC and CL for lossless transmission lines (though the diagonalizability of their complex analogues, ZY and YZ matrices, is not guaranteed for lossy lines), and demonstrate the properties of their eigenvalues. We have developed an algorithm to decouple one type of matrix differential equation, and to construct the characteristic impedance matrix Z/sub c/ explicitly and efficiently. Based on this work, the congruence and similarity transformations, which have caused considerable confusion and not a few errors in the decoupling and solution of the matrix telegrapher's equations, are analyzed and summarized. In addition, we also demonstrate that under certain conditions, the diagonalization of two or more matrices by means of the congruence or similarity transformations may lead to coordinate system "mismatch" and introduce erroneous results. >

Approximation of isolated eigenvalues of general singular ordinary differential operators

Abstract: Let A be a self-adjoint operator defined by a general singular ordinary differential expression τ on an interval (a, b), − ∞ ≤ a < b ≤ ∞. We show that isolated eigenvalues in any gap of the essential spectrum of A are exactly the limits of eigenvalues of suitably chosen self-adjoint realizations An of τ on subintervals (an, bn) of (a, b) with an → a, bn → b. This means that eigenvalues of singular ordinary differential operators can be approximated by eigenvalues of regular operators.

Probability of an Eigenvalue Number Fluctuation in an Interval of a Random Matrix Spectrum.

Abstract: We calculate the probability to find exactly $n$ eigenvalues in a spectral interval of a large random $N \times N$ matrix when this interval contains $s \ll N$ eigenvalues on average. The calculations exploit an analogy to the problem of finding a two-dimensional charge distribution on the interface of a semiconductor heterostructure under the influence of a split gate.

Adaptive multigrid techniques for large-scale eigenvalue problems: Solutions of the Schrödinger problem in two and three dimensions.

Abstract: Multigrid (MG) algorithms for large-scale eigenvalue problems (EP), obtained from discretizations of partial differential EP, have often been shown to be more efficient than single level eigenvalue algorithms. This paper describes a robust and efficient, adaptive MG eigenvalue algorithm. The robustness of the present approach is a result of a combination of MG techniques introduced here, i.e., the completion of clusters; the adaptive treatment of clusters; the simultaneous treatment of solutions in each cluster; the miltigrid projection (MGP) coupled with backrotations; and robustness tests. Due to the MGP, the algorithm achieves a better computational complexity and better convergence rates than previous MG eigenvalue algorithms that use only fine level projections. These techniques overcome major computational difficulties related to equal and closely clustered eigenvalues. Some of these difficulties were not treated in previous MG algorithms. Computational examples for the Schr\"odinger eigenvalue problem in two and three dimensions are demonstrated for cases of special computational difficulties, which are due to equal and closely clustered eigenvalues. For these cases, the algorithm requires O(qN) operations for the calculation of q eigenvectors of size N, using a second order approximation. The total computational cost is equivalent to only a few Gause-Seidel relaxations per eigenvector.

Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system

Abstract: The author extends the rigorous Chandezon formalism (1982) to accommodate three-dimensional modulation profiles. The basic feature lies in the use of a coordinate system that maps the interface onto a plane. As in the two-dimensional case, one is led to a linear system of differential equations whose solution is obtained through the calculation of the eigenvalues and eigenvectors of a matrix dependent on the geometry and on the index of the medium. Results are compared with those obtained by the methods of others.

Non-Symmetric Matrix Riccati Equations

Abstract: We prove a fundamental representation formula for all solutions of the matrix Riccati differential equation and of the corresponding algebraic Riccati equation. This formula contains the complete information on the phase portrait of the matrix equation and on the structure of the set F of all solutions of the corresponding algebraic equation. In particular we describe all constant, periodic and almost-periodic solutions of the matrix Riccati differential equation. Further we give an application of the fundamental representation formula to the investigation of non-autonomous Riccati equations.

Inverse scattering problem for anisotropic media

Abstract: An inverse scattering problem for a second order matrix differential equation on the line related to the wave propagation in anisotropic media is studied herein. A reconstruction procedure is given based on the Riemann–Hilbert problem of analytic factorization of matrix functions and a uniqueness theorem is proven.

Eigenvalues and Eigenvectors

Abstract: This chapter discusses the concepts eigenvalues and eigenvectors. For any square matrix A over F , the polynomial det( A — xI ) is the characteristic polynomial of A . The equation det( A — xI ) = 0 is the characteristic equation of A , and the solutions of det( A — xI ) = 0 are called the eigenvalues of A . The set of all eigenvalues of A is called the spectrum of A . In the chapter, T denotes a field, V denotes a finite-dimensional vector space over T , and T denotes a linear operator on V . The chapter discusses the important connection between the eigenvalues of linear operators and those of matrices. If the n × n matrix A represents T relative to the basis A of V , then X is an eigenvector of A corresponding to λ only if X is the coordinate matrix relative to A of an eigenvector of T corresponding to the same eigenvalue.

Differential equations and factorization property for the one-dimensional Schrodinger equation with position-dependent mass

Abstract: The variable phase method is applied to the one dimensional Schrodinger equation with position-dependent (effective) mass, to derive first-order differential equations for the transmission and reflection amplitudes, and bound-state energies, which are particularly convenient for numerical computations. When the mass and potential have the same asymptotics at both ends of the real line, the method also allows to prove a factorization property of the scattering matrix.

Homotopy method for the numerical solution of the eigenvalue problem of self-adjoint partial differential operators

Abstract: GivenA1, the discrete approximation of a linear self-adjoint partial differential operator, the smallest few eigenvalues and eigenvectors ofA1 are computed by the homotopy (continuation) method. The idea of the method is very simple. From some initial operatorA0 with known eigenvalues and eigenvectors, define the homotopyH(t)=(1−t)A0+tA1, 0≤t≤1. If the eigenvectors ofH(t0) are known, then they are used to determine the eigenpairs ofH(t0+dt) via the Rayleigh quotient iteration, for some value ofdt. This is repeated untilt becomes 1, when the solution to the original problem is found. A fundamental problem is the selection of the step sizedt. A simple criterion to selectdt is given. It is shown that the iterative solver used to find the eigenvector at each step can be stabilized by applying a low-rank perturbation to the relevant matrix. By carrying out a small part of the calculation in higher precision, it is demonstrated that eigenvectors corresponding to clustered eigenvalues can be computed to high accuracy. Some numerical results for the Schrodinger eigenvalue problem are given. This algorithm will also be used to compute the bifurcation point of a parametrized partial differential equation.

Computing continuous numerical solutions of matrix differential equations

Abstract: In this paper, we construct analytical approximate solutions of initial value problems for the matrix differential equation X ′( t ) = A ( t ) X ( t ) + X ( t ) B ( t ) + L ( t ), with twice continuously differentiable functions A ( t ), B ( t ), and L ( t ), continuous. We determine, in terms of the data, the existence interval of the problem. Given an admissible error e, we construct an approximate solution whose error is smaller than e uniformly, in all the domain.

Asymptotic behavior of the self-defocusing nonlinear Schrödinger equation for piecewise constant initial conditions

Abstract: In this paper we use a transfer matrix method to calculate the asymptotic behavior of the nonlinear Schrodinger (NLS) equation in a self-defocusing medium for piecewise constant initial conditions. Treating initial conditions that consist of m repeated regions, we show that the eigenvalues associated with this problem appear in bands, and, as m tends to infinity, we obtain the eigenvalue density of states for these bands. Comparing results from the transfer matrix approach to the results for a Bloch function method, we show that the edges of a region with periodic initial conditions result in a finite number of additional eigenvalues that appear outside the bands.

Calculating the eigenvalues of ordinary differential equations

Abstract: Numerical algorithms without saturation for calculating the eigenvalues and eigenfunctions of ordinary differential equations with smooth coefficients are considered. The results ofdifferent methods are discussed and an error estimate is given.

Eigenvalues of matrix commutators

Abstract: We characterize the eigenvalues of [X,A]=XA−AX, where A is an n by n fixed matrix and X runs over the set of the matrices of the same size.

Isospectral matrix flows for numerical analysis

Abstract: Publisher Summary Current interest in analog computation and neural networks has motivated the investigation of matrix eigenvalue problems via eigenvalue preserving differential equations. An example is Brockett's recent work on a double Lie bracket quadratic matrix differential equation with applications to matrix diagonalization, sorting and linear programming. Another example is the Toda flow as well as many other classical completely integrable Hamiltonian systems. Such isospectral flows appear to be a useful tool for solving matrix eigenvahe problems. Moreover, generalizations of such flows are able to compute the singular value decomposition of arbitrary rectangular matrices. In neural network theory similar flows have appeared in investigations on learning dynamics for networks achieving the principal component analysis. This chapter attempts to survey these recent developments, with special emphasis on flows solving matrix eigenvalue problems. Discretizations of the flows based on geodesic approximations lead to new, although slowly convergent, algorithms. A modification of such schemes leads to a new Jacobi type algorithm, which is shown to be quadratically convergent.

Three-Dimensional Air-to-Air Missile Trajectory Shaping

Abstract: Open-loop range/energy/time optimal trajectories are synthesized for a three-dimensional point-mass model of a boost-sustain-coast air-to-air missile (AAM) in atmospheric flight. Boundaries of attainable sets for various fixed flight times are obtained by numerically solving the associated two-point boundary value problem (TPBVP) implied by the first-order necessary conditions of optimal control. The attainable set provides insight into the range capabilities of the missile in all directions in the down range-cross range plane for prescribed end-game energy requirements. Necessary and sufficient conditions for optimality are checked for candidate extremals. A new governing matrix differential equation and suitable boundary conditions are derived to check for conjugate points along regular or regularized extremals. This new matrix differential equation overcomes the difficulty of standard matrix Riccati equations for conjugate point testing, which have components whose values go to infinity even in the absence of conjugate points. The open-loop solutions presented in this study can be utilized to generate a neighboring near-optimal guidance scheme.

The Jacobi-Davidson method for eigenvalue problems as an accelerated inexact Newton scheme

Abstract: We discuss a new method for the iterative computation of a portion of the spectrum of a large sparse matrix. The matrix may be complex and non-normal. The method also delivers the Schur vectors associated with the computed eigenvalues. The eigenvectors can easily be computed from the Schur vectors, and for stability reasons we prefer the approach with Schur vectors. The method is based on the recently introduced Jacobi-Davidson algorithm [16]. This method improves the Davidson method and its generalizations. We also show how the Davidson's methods, including the new one, can be viewed as accelerated inexact Newton schemes.

Theory of quasi-flute modes in toroidal plasma configurations

Abstract: The problem of quasi-flute mode stability in toroidal plasma configuration is reduced to an ordinary differential equation of the second order. This equation has the same form as the "distilled" ballooning equation, however, it is derived in Fourier space of wavenumbers, where the ratio of the radial component of the wavevector to the poloidal one is an independent variable, instead of using ballooning variables. The obtained equation is transformed to the form of a stationary Schrodinger equation. It is shown that the discrete eigenvalue spectrum of this equation corresponds to the unstable modes. Two regimes of quasi-flute mode excitation corresponding to the cases of fulfilment and violation of the Mercier criterion of plasma stability are found. The dispersion relations defining quasi-flute perturbation growth rates are derived for both regimes.

Factorization of selfadjoint quadratic matrix polynomials with real spectrum

Abstract: Factorization theorems, and properties of sets of eigenvectors, are established for regular selfadjoint quatratic matrix polynomials L(λ) whose leading coefficeint is indefinite or possibly singular, and for which all eigenvalues are real of definite type. The two linear factors obtained have spectra which are just the eigenvalues of L(λ) of positive and negative types, respectively.

Universal behavior of correlations between eigenvalues of random matrices

Abstract: The universal connected correlations proposed recently between eigenvalues of unitary random matrices is examined numerically. We perform an ensemble average by the Monte Carlo sampling. Although density of eigenvalues and a bare correlation of the eigenvalues are not universal, the connected correlation shows a universal behavior after smoothing.

Construction of analogs of the Lyapunov equation for a matrix polynomial

Abstract: We develop a method for localization of the eigenvalues of a matrix polynomial. This method is related to a generalization and solution of the Lyapunov equation.

Localization of Eigenvalues in Elliptic Regions

Abstract: We prove that the eigenvalues λ j of an n by n complex matrix A with its characteristic polynomial having real coefficients lie in the elliptic region defined by β 2 χ− tr A n 2 +α 2 y 2 ≤α 2 β 2 , where α n−1 n ∑ n=1 n ( Re λ k ) 2 − ( Re(te A )) 2 n 1 2 and β= n−1 n ∑ k=1 n ( Im λ k ) 2 1 2 This region is intersected with the strip |y|≤ 1 2 ∑ k=1 u Im λ k ) 2 1 2 to obtain an improved eigenvalue localization region. We also give bounds for the semiaxes, which can be computed without knowing the eigenvalues of A . When A has r n nonzero eigenvalues, we obtain a smaller elliptic region containing such nonzero eigenvalues.

Performances analysis of parameterized adaptive eigenspace algorithms

Abstract: We address adaptive estimation methods of eigenspaces of covariance matrices. We are interested in methods based on several coupled maximizations or minimizations of Rayleigh ratios where the constraints are replaced by appropriate parameterizations (Givens and mixed Givens/Householder). We prove the convergence of these algorithms with the help of the associated ordinary differential equation, and we propose an evaluation of the performance by computing the variances of the estimated eigenvectors for fixed gain factors. We show that these variances are very sensitive to the difference between two consecutive eigenvalues. Moreover, they also depend on whether the successive analyzed vector signals are correlated or not, and thus greatly depend on the origin of the covariance matrices of interest (spatial, temporal, spatio-temporal). Finally we show that the performance can be improved when the centro-symmetric property of some of those covariance matrices is taken into account.