Bounds for Polynomial’s Roots from Fiedler and Sparse Companion Matrices for Submultiplicative Matrix Norms

We use submultiplicative companion matrix norms to provide new bounds for roots for a given polynomial P(X) over the field C[X]. From a n×n Fiedler companion matrix C, sparse companion matrices and triangular Hessenberg matrices are introduced. Then, we identify a special triangular Hessenberg matrix Lr, supposed to provide a good estimation of the roots. By application of Gershgorin’s theorems to this special matrix in case of submultiplicative matrix norms, some estimations of bounds for roots are made. The obtained bounds have been compared to known ones from the literature precisely Cauchy’s bounds, Montel’s bounds and Carmichel-Mason’s bounds. According to the starting formel of Lr, we see that the more we have coefficients closed to zero with a norm less than 1, the more the Sparse method is useful.

STABILITY ANALYSIS FOR LINEAR DELAY DIFFERENTIAL EQUATIONS AND NUMERICAL EXAMPLES

The asymptotic stability of delay differential equation x′(t)=Ax(t)+Bx(t τ) is concerned with,where A,B∈C d×d are constant complex matrices, x(t τ)=(x 1(t-τ 1),x 2(t-τ 2),...,x d(t-τ d))T,τ k>0(k=1,...,d) stand for constant delays. Two criteria through evaluation of a harmonic function on the boundary of a certain region are obtained. The similar results for neutral delay differential equation x′(t)=Lx(t)+Mx(t-τ)+Nx′(t-τ) are also obtained,where L,M and N∈C d×d are constant complex matrices and τ>0 stands for constant delay. Numerical examples are showed to check the results which are more general than those already reported.

ON L^p-MATRICIALLY NORMED SPACES

It is proved that there is only one L^P-matricially normed space of dimension 1 and that quotient spaces of L^P-matricially normed spaces are also L^P-matricially normed spaces. Some properties of L^P-matricially normed spaces are given.

Norm Properties of Y-numerical Radii

Given an n×n complex matrix A and an n-dimensional complex vector y=(ν1 , ··· , νn ), the y-numerical radius of A is the nonnegative quantity ry(A)=max{n∑j=1ν*jAx︱:Axj︱: x*jxj=1,xj ∈Cn}.Here Cn is an n-dimensional linear space overthe complex field C. For y = (1, 0, ··· , 0) it reduces to the classical radius r(A) =max {|x*Ax|: x*x=1}.We show that ry is a generalized matrix norm if and only ifn∑j=1νj≠ 0.Next, we study some properties of the y-numerical radius of matrices andvectors with non-negative entries.

Direction of arrival estimation method based on quantum electromagnetic field optimization in the impulse noise

In order to resolve direction finding problems in the impulse noise,a direction of arrival(DOA)estimation method is proposed.The proposed DOA estimation method can restrain the impulse noise by using infinite norm exponential kernel covariance matrix and obtain excellent performance via the maximumlikelihood(ML)algorithm.In order to obtain the global optimal solutions of this method,a quantum electromagnetic field optimization(QEFO)algorithm is designed.In view of the QEFO algorithm,the proposed method can resolve the difficulties of DOA estimation in the impulse noise.Comparing with some traditional DOA estimation methods,the proposed DOA estimation method shows high superiority and robustness for determining the DOA of independent and coherent sources,which has been verified via the Monte-Carlo experiments of different schemes,especially in the case of snapshot deficiency,low generalized signal to noise ratio(GSNR)and strong impulse noise.Beyond that,the Cramer-Rao bound(CRB)of angle estimation in the impulse noise and the proof of the convergence of the QEFO algorithm are provided in this paper.

On y-numerical Radius and Its Stability

Let A be an n×n complex matrix,and let y =(α1,...,αn) be an n-dimensional complex vector.The y-numerical radius of A,denoted by ry(A),is defined follows:r_y(A) = max {|sum form i=1 to n α_ix_i~* Ax_i|:x_i~* x_i = 1,x_i ∈ C^n},where Cn is an n-dimensional vector space over complex field C.In this paper we studynorm properties and stability of y-numerical radius.

Least-square Solutions of Inverse Problems for Anti-symmetric and Skew-symmetric Matrices

The least-square solutions of inverse problem for anti-symmetric and skew-symmetric matrices are studied. In addition, the problem of using anti-symmetric and skew-symmetric matrices to construct the optimal approximation to a given matrix is discussed, the necessary and sufficient conditions for the problem are derived, and the expression of the solution is provided. A numerical example is given to show the effectiveness of the proposed method.

STABLE DOUBLE LR ALGORITHM AND ITS ERROR ANALYSIS

In this paper,the normative matrices and their double LR transformationwith origin shifts are defined,and the essential relationship between the double LR transformation of a normative matrix and the QR transformation of the related symmetrictridiagonal matrix is proved.We obtain a stable double LR algorithm for double LRtransformation of normative matrices and give the error analysis of our algorithm.Theoperation number of the stable double LR algorithm for normative matrices is only foursevenths of the rational QR algorithm for real symmetric tridiagonal matrices.

Harmonic Data Recovery Method Based on Multivariate Norm Matrix

During state perception of a power system, fragments of harmonic data are inevitably lost owing to the loss of synchronization signals, transmission delays, instrument failures, or other factors. A harmonic data recovery method is proposed based on multivariate norm matrix in this paper. The proposed method involves dynamic time warping for correlation analysis of harmonic data, normalized cuts for correlation clustering of power-quality monitoring devices, and adaptive alternating direction method of multipliers for multivariable norm joint optimization. Compared with existing data recovery methods, our proposed method maintains excellent recovery accuracy without requiring prior information or optimization of the power-quality monitoring device. Simulation results on the IEEE 39-bus and IEEE 118-bus test systems demonstrate the low computational complexity of the proposed method and its robustness against noise. In addition, the application of the proposed method to field data from a real-world system provides consistent results with those obtained from simulations.

EQUISTABILITY OF THE MATRIX DIFFERENTIAL EQUATIONS

In this paper the problem about equistability of the matrix equistability of the matrix differential equations have been discussed, and some criteria for equistability have been given.

Least-Squares Solution of Inverse Problem for Hermitian Anti-reflexive Matrices and Its Appoximation

In this paper, we first consider the least-squares solution of the matrix inverse problem as follows： Find a hermitian anti-reflexive matrix corresponding to a given generalized reflection matrix J such that for given matrices X, B we have minA ｜｜AX - B｜｜. The existence theorems are obtained, and a general representation of such a matrix is presented. We denote the set of such matrices by SE. Then the matrix nearness problem for the matrix inverse problem is discussed. That is： Given an arbitrary A^＊, find a matrix A E SE which is nearest to A^＊ in Frobenius norm. We show that the nearest matrix is unique and provide an expression for this nearest matrix.