Perturbation of monic matrix polynomials

Matrix polynomials of the form $P(\lambda)=I\lambda^2+A_1\lambda+A_0$ (where A0 and A1 are n × n Hermitian matrices and $\lambda$ is a complex variable) arise in many applications. The numerical range of such a polynomial is \[ W(P) = \{\lambda \in \mathbb{C} : x^*P(\lambda)x = 0 \mbox{ for some nonzero } x \in \mathbb{C}^n \} \] and it always contains the spectrum of $P(\lambda)$, i.e., the set of zeros of $\textup{det}P(\lambda)$. Properties of the numerical range are developed in detail, taking advantage of the close connection between W(P) and the classical numerical range (field of values) of the (general) complex matrix A:=A0+iA1. Eigenvalues and nondifferentiable points on the boundary are examined and a procedure for the numerical determination of W(P) is presented and used for several illustrations. Some extensions of the theory to more general polynomials $P(\lambda)$ are also discussed, as well as special cases describing vibrating systems.

The numerical range of a selfadjoint matrix polynomial \( L\left( \lambda \right) = \sum\nolimits_j^\ell { = 0{\lambda ^j}{A_j}} \) is the set of points μ∈ℂ for which x* L(μ)x = 0 for some nonzero vector x. As for the classical eigenvalue problem (when L(λ) = λ I − A), the spectrum of L(λ) is contained in its numerical range. Properties of the numerical range are investigated with special emphasis on the cases when L(λ) has only real spectrum (and, possibly, the point at infinity) and when the coefficients of the matrix polynomial are real symmetric matrices.

Comparing the performance of a nested to a continuous evolution strategy with covariance matrix adaption for optimization of drilling EDM

The family of q-Konhauser matrix polynomials have been extended to Konhauser matrix polynomials. The purpose of the present work is to show that an extension of the explicit forms, generating matrix functions, matrix recurrence relations and Rodrigues-type formula for these matrix polynomials are given, our desired results have been established and their applications are presented.

Two sufficient conditions under which the roots of the determinant of a given ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m \times m</tex> ) matrix polynomial of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th order lie in the open left-half plane have been obtained. The first condition is given in terms of the positive definiteness of an ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mn \times mn</tex> ) symmetric matrix, while the second condition is given in terms of the positive definiteness of an ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m \times m</tex> ) matrix that is a function of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> , Re <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s \leq 0</tex> . These conditions are represented in terms of rational functions of the coefficient matrices of the given matrix polynomial. Therefore, the explicit computation of the determinant polynomial is not required.

On the quasi-stability criteria of monic matrix polynomials

A polynomial p is said to be Γ-stable if all its roots lie within a given domain Γ in the complex plane. The Γ-stability of an entire family of polynomials, defined by selecting the coefficients of p from specified complex sets, can be verified by (i) testing the Γ-stability of a single member, and (ii) checking that the “total value set” V * for p along the domain boundary ∂Γ does not contain 0 (V * is defined as the set of all values of p for each point on ∂Γ and every possible choice of the coefficients). The methods of Minkowski geometric algebra —the algebra of point sets in the complex plane — offer a natural language for the stability analysis of families of complex polynomials. These methods are introduced, and applied to analyzing the stability of disk polynomials with coefficients selected from circular disks in the complex plane. In this context, V * may be characterized as the union of a one-parameter family of disks, and we show that the Γ-stability of a disk polynomial can be verified by a finite algorithm (a counterpart to the Kharitonov conditions for rectangular coefficient sets) that entails checking that at most two real polynomials remain positive for all t, when the domain boundary ∂Γ is a given polynomial curve γ(t).Furthermore, the “robustness margin” can be determined by computing the real roots of a real polynomial.

Families of matrices smoothly depending on a vector of parameters are considered. Arnold [Russian Math. Surveys, 26 (1971), pp. 29--43] and Galin [Uspekhi Mat. Nauk, 27 (1972), pp. 241--242] have found and listed normal forms of families of complex and real matrices (miniversal deformations), to which any family of matrices can be transformed in the vicinity of a point in the parameter space by a change of basis, smoothly dependent on a vector of parameters, and by a smooth change of parameters. In this paper a constructive method of determining functions describing a change of basis and a change of parameters, transforming an arbitrary family to the miniversal deformation, is suggested. Derivatives of these functions with respect to parameters are determined from a recurrent procedure using derivatives of the functions of lower orders and derivatives of the family of matrices. Then the functions are found as Taylor series. Examples are given. The suggested method allows using efficiently miniversal deformations for investigation of different properties of matrix families. This is shown in the paper where tangent cones (linear approximations) to the stability domain at the singular boundary points are found.

State estimation aims at approximately reconstructing the solution $u$ to a parametrized partial differential equation from $m$ linear measurements when the parameter vector $y$ is unknown. Fast numerical recovery methods have been proposed in Maday et al. [Internat. J. Numer. Methods Engrg., 102 (2015), pp. 933--965] based on reduced models which are linear spaces of moderate dimension $n$ that are tailored to approximate the solution manifold ${\cal M}$ where the solution sits. These methods can be viewed as deterministic counterparts to Bayesian estimation approaches and are proved to be optimal when the prior is expressed by approximability of the solution with respect to the reduced model [P. Binev et al., SIAM/ASA J. Uncertain. Quantif., 5 (2017), pp. 1--29]. However, they are inherently limited by their linear nature, which bounds from below their best possible performance by the Kolmogorov width $d_m({\cal M})$ of the solution manifold. In this paper, we propose to break this barrier by using simple nonlinear reduced models that consist of a finite union of linear spaces $V_k$, each having dimension at most $m$ and leading to different estimators $u_k^*$. A model selection mechanism based on minimizing the PDE residual over the parameter space is used to select from this collection the final estimator $u^*$. Our analysis shows that $u^*$ meets optimal recovery benchmarks that are inherent to the solution manifold and not tied to its Kolmogorov width. The residual minimization procedure is computationally simple in the relevant case of affine parameter dependence in the PDE. In addition, it results in an estimator $y^*$ for the unknown parameter vector. In this setting, we also discuss an alternating minimization (coordinate descent) algorithm for joint state and parameter estimation that potentially improves the quality of both estimators.

On factorization of matrix polynomials

We study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex‐valued weight function, , over the interval , where is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, , due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as have recently been studied for , and our main goal is to extend these results to all in the complex plane. We first use the technique of continuation in parameter space, developed in the context of the theory of integrable systems, to extend previous results on the so‐called modified external field from the imaginary axis to the complex plane minus a set of critical curves, called breaking curves. We then apply the powerful method of nonlinear steepest descent for oscillatory Riemann–Hilbert problems developed by Deift and Zhou in the 1990s to obtain asymptotics of the recurrence coefficients of these polynomials when the parameter is away from the breaking curves. We then provide the analysis of the recurrence coefficients when the parameter approaches a breaking curve, by considering double scaling limits as approaches these points. We see a qualitative difference in the behavior of the recurrence coefficients, depending on whether or not we are approaching the points or some other points on the breaking curve.

The parametric minimax problem, which finds the parameter value minimizing the weight of a solution of a combinatorial maximization problem, is a fundamental problem in sensitivity analysis. Moreover, several problems in computational geometry can be formulated as parametric minimax problems. The parametric search paradigm gives an efficient sequential algorithm for a convex parametric minimax problem with one parameter if the original non-parametric problem has an efficient parallel algorithm. We consider the parametric minimax problem with d parameters for a constant d, and solve it by using multidimensional version of the parametric search paradigm. As a new feature, we give a feasible region in the parameter space in which the parameter vector must be located.Typical results obtained as applications are: (1) Efficient solutions for some geometric problems, including theoretically efficient solutions for the minimum diameter bridging problem in d-dimensional space between convex polytopes. (2) Parametric polymatroid optimization, for example, O(n log n) time algorithm to compute the parameter vector minimizing k-largest linear parametric elements with d dimensions.

Stability analysis of high-order Hopfield type neural networks with uncertainty

Multistationarity in the activation of a MAPK: Parametrizing the relevant region in parameter space

This paper presents a system for leather inspection based upon visual textural properties of the material surface. Defects are isolated from the complex and not homogeneous background by analyzing their strongly oriented structure. The patterns to be analyzed are represented in an appropriate parameter space using an optimization approach: in this way a parameter vector is associated to each different textured region in the original image. Finally a filter process, based upon knowledge about the parameter vectors representing the leather without defects, detects and classifies any abnormality.Subject termsleather inspectiontexture analysistexture classificationoriented structuresparameter spacevectorial bases