A unified framework for the numerical solution of general quadratic matrix equations

^** Email: vassilios.tsachouridis{at}ieee.org^*** Email: N.karcanias{at}city.ac.uk^**** Email: ixp{at}le.ac.uk Algebraic quadratic equations are special cases of a singlegeneralized algebraic quadratic matrix equation (GQME). Thispaper focuses on the numerical solution of the GQME using probability-1homotopy methods. A synoptic review of these methods and theirapplication to algebraic matrix equations is provided as background.A large variety of analysis and design problems in systems andcontrol are reported as special cases of the presented frameworkand some of them are illustrated via numerical examples fromthe literature.     

Solvability of quadratic matrix equations

Solvability conditions are studied in this paper for a quadratic matrix Riccati equation arising in studies of the Chapman-Enskog projection for a Cauchy problem and a mixed problem for momentum approximations of kinetic equations. The structure of the matrix equation permits one to formulate necessary and sufficient solvability conditions in terms of eigenvectors and associated vectors for the matrix composed from the coefficients.     

The homogeneous projective transformation of general quadratic matrix equations

^** Email: vassilios.tsachouridis{at}ieee.org^*** Email: basil.kouvaritakis{at}eng.ox.ac.uk Algebraic quadratic equations are a special case of a singlegeneralized algebraic quadratic matrix equation (GQME). Hence,the importance of that equation in science and engineering isevident. This paper focus on the study of solutions of thatGQME and a unified framework for the characterization and identificationof solutions at infinity and of finite solutions of generalquadratic algebraic matrix equations is presented. The analysisis based on the concept of homogeneous projective transformationfor general polynomial systems (Morgan, 1986). In addition,a numerical error analysis for the computed solutions is providedfor the assessment of numerical accuracy, stability and conditioningof the computed solutions. The proposed framework is independentof any numerical method and therefore it can be used along withvarious possible numerical methods for the GQME solution, especiallymatrix flow-based algorithms (Chu, 1994) (e.g. continuation/homotopy,Morgan, 1989).     

Error Analysis of the Symplectic Lanczos Method for the Symplectic Eigenvalue Problem

A rounding error analysis of the symplectic Lanczos algorithm for the symplectic eigenvalue problem is given. It is applicable when no break down occurs and shows that the restriction of preserving the symplectic structure does not destroy the characteristic feature of nonsymmetric Lanczos processes. An analog of Paige's theory on the relationship between the loss of orthogonality among the Lanczos vectors and the convergence of Ritz values in the symmetric Lanczos algorithm is discussed. As to be expected, it follows that (under certain assumptions) the computed J-orthogonal Lanczos vectors loose J-orthogonality when some Ritz values begin to converge.     

The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems

In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous, periodic, and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems, and unify known results under various convexity conditions.

     

General nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations

The general nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations is considered. A method is proposed for reducing the problem to one for a Hamiltonian system. Results for Hamiltonian systems previously obtained by the authors are extended to this system.     

Hessenberg matrix for sums of Hermitian positive definite matrices and weighted shifts

In this work, we introduce an algebraic operation between bounded Hessenberg matrices and we analyze some of its properties. We call this operation m-sum and we obtain an expression for it that involves the Cholesky factorization of the corresponding Hermitian positive definite matrices associated with the Hessenberg components.This work extends a method to obtain the Hessenberg matrix of the sum of measures from the Hessenberg matrices of the individual measures, introduced recently by the authors for subnormal matrices, to matrices which are not necessarily subnormal.Moreover, we give some examples and we obtain the explicit formula for the m-sum of a weighted shift. In particular, we construct an interesting example: a subnormal Hessenberg matrix obtained as the m-sum of two not subnormal Hessenberg matrices.     

Numerical analysis for the quadratic matrix equations from a modification of fixed‐point type

In this paper, we study the quadratic matrix equations. To improve the application of iterative schemes, we use a transform of the quadratic matrix equation into an equivalent fixed‐point equation. Then, we consider an iterative process of Chebyshev‐type to solve this equation. We prove that this iterative scheme is more efficient than Newton's method. Moreover, we obtain a local convergence result for this iterative scheme. We finish showing, by an application to noisy Wiener‐Hopf problems, that the iterative process considered is computationally more efficient than Newton's method.     

Isolated solutions of quadratic matrix equations

We study the solutions X to the quadratic operator equation XBX+XA?DX?C=0 via the invariant subspace structure of an associated operator matrix. We translate a theorem of R.G. Douglas and C. Pearcy into a characterization of the isolated solutions to the matrix equation XBX+XA?DX?C=0, and we discuss conditions which ensure that the equation has a unique solution.     

Transforming algebraic Riccati equations into unilateral quadratic matrix equations

The problem of reducing an algebraic Riccati equation XCX − AX − XD + B = 0 to a unilateral quadratic matrix equation (UQME) of the kind PX ² + QX + R = 0 is analyzed. New transformations are introduced which enable one to prove some theoretical and computational properties. In particular we show that the structure preserving doubling algorithm (SDA) of Anderson (Int J Control 28(2):295–306, 1978) is in fact the cyclic reduction algorithm of Hockney (J Assoc Comput Mach 12:95–113, 1965) and Buzbee et al. (SIAM J Numer Anal 7:627–656, 1970), applied to a suitable UQME. A new algorithm obtained by complementing our transformations with the shrink-and-shift technique of Ramaswami is presented. The new algorithm is accurate and much faster than SDA when applied to some examples concerning fluid queue models.     

Low-rank updates and divide-and-conquer methods for quadratic matrix equations

Numerical Algorithms - In this work, we consider two types of large-scale quadratic matrix equations: continuous-time algebraic Riccati equations, which play a central role in optimal and robust...     

Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays

A symmetrizer of a nonsymmetric matrix A is the symmetric matrixX that satisfies the equationXA =A ^tX, wheret indicates the transpose. A symmetrizer is useful in converting a nonsymmetric eigenvalue problem into a symmetric one which is relatively easy to solve and finds applications in stability problems in control theory and in the study of general matrices. Three designs based on VLSI parallel processor arrays are presented to compute a symmetrizer of a lower Hessenberg matrix. Their scope is discussed. The first one is the Leiserson systolic design while the remaining two, viz., the double pipe design and the fitted diagonal design are the derived versions of the first design with improved performance.     

A matrix approach for constructing quadratic APN functions

A one to one correspondence is given between quadratic homogeneous APN functions and a special kind of matrices which we call as QAM’s. By modifying the elements of a known QAM, new quadratic APN functions can be constructed. Based on the nice mathematical structures of the QAM’s, an efficient algorithm for constructing quadratic APN functions is proposed. On \(\mathbb {F}_{2^7}\) , we have found 471 new CCZ-inequivalent quadratic APN functions, which is 20 times more than the number of the previously known ones. Before this paper, It is only found 23 classes of CCZ-inequivalent APN functions on \(\mathbb {F}_{2^8}\) . With the method of this paper, we have found 2,252 new CCZ-inequivalent quadratic APN functions, and this number is still increasing.     

On the numerical solution of stochastic quadratic integral equations via operational matrix method

In this paper, stochastic operational matrix of integration based on delta functions is applied to obtain the numerical solution of linear and nonlinear stochastic quadratic integral equations (SQIEs) that appear in modelling of many real problems. An important advantage of this method is that it dose not need any integration to compute the constant coefficients. Also, this method can be utilized to solve both linear and nonlinear problems. By using stochastic operational matrix of integration together collocation points, solving linear and nonlinear SQIEs converts to solve a nonlinear system of algebraic equations, which can be solved by using Newton's numerical method. Moreover, the error analysis is established by using some theorems. Also, it is proved that the rate of convergence of the suggested method is O(h²). Finally, this method is applied to solve some illustrative examples including linear and nonlinear SQIEs. Numerical experiments confirm the good accuracy and efficiency of the proposed method.     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

On generalized hamiltonian systems

1. PreliminaryIt is well known that{1] a 8ymPlectic form is invariant along the trajectory of a Hamilto-nian system. Based on this fundamental property, certain techniques have been developed.The purpose of this paper is to extend such an approach to a wider class of dynamic systeIns,namely, genera1ized Hamiltonian systems. Our purpose is to investigate a class of dynaInicsystems, which possess a certain "geometric structure".Deflnition 1.1[1'2]. Let M be a tIlallifo1d. w E fl'(M) is call…     

线性矩阵哈密顿系统的振动性

研究了线性矩阵 Hamilton系统X′=A( t) X + B( t) YY′=C( t) X -A*( t) Y　　 t≥ 0的振动性 .其中 A( t) ,B( t) ,C( t) ,X,Y为实 n× n矩阵值函数 ,B,C为对称矩阵 ,B正定 .借助于正线性泛函 ,采用加权平均法 ,得到了该系统的非平凡预备解的振动性 .这些结果推广、改进了许多已知的结果     

Vector space of linearizations for the quadratic two-parameter matrix polynomial

Given a quadratic two-parameter matrix polynomial Q(λ,?μ), we develop a systematic approach to generating a vector space of linear two-parameter matrix polynomials. The purpose for constructing this vector space is that potential linearizations of Q(λ,?μ) lie in it. Then, we identify a set of linearizations and describe their constructions. Finally, we determine a class of linearizations for a quadratic two-parameter eigenvalue problem.