Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise

We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of multivariate ARMA equations with independent and identically distributed driving noise. For general ARMA(p, q) equations these conditions are expressed in terms of the coefficient polynomials of the defining equations and moments of the driving noise sequence, while for p =?1 an additional characterization is obtained in terms of the Jordan canonical decomposition of the autoregressive matrix, the moving average coefficient matrices and the noise sequence. No a priori assumptions are made on either the driving noise sequence or the coefficient matrices.     

Sine‐cosine wavelet method for fractional oscillator equations

Purpose In this article, a novel computational method is introduced for solving the fractional nonlinear oscillator differential equations on the semi‐infinite domain. The purpose of the proposed method is to get better and more accurate results. Design/methodology/approach The proposed method is the combination of the sine‐cosine wavelets and Picard technique. The operational matrices of fractional‐order integration for sine‐cosine wavelets are derived and constructed. Picard technique is used to convert the fractional nonlinear oscillator equations into a sequence of discrete fractional linear differential equations. Operational matrices of sine‐cosine wavelets are utilized to transformed the obtained sequence of discrete equations into the systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear oscillator equations. Findings The convergence and supporting analysis of the method are investigated. The operational matrices contains many zero entries, which lead to the high efficiency of the method, and reasonable accuracy is achieved even with less number of collocation points. Our results are in good agreement with exact solutions and more accurate as compared with homotopy perturbation method, variational iteration method, and Adomian decomposition method. Originality/value Many engineers can utilize the presented method for solving their nonlinear fractional models.     

BANDED TOEPLITZ PRECONDITIONERS FOR TOEPLITZ MATRICES FROM SINC METHODS

We give general expressions, analyze algebraic properties and derive eigenvalue bounds for a sequence of Toeplitz matrices associated with the sinc discretizations of various orders of differential operators. We demonstrate that these Toeplitz matrices can be satisfactorily preconditioned by certain banded Toeplitz matrices through showing that the spectra of the preconditioned matrices are uniformly bounded. In particular, we also derive eigenvalue bounds for the banded Toeplitz preconditioners. These results are elementary in constructing high-quality structured preconditioners for the systems of linear equations arising from the sinc discretizations of ordinary and partial differential equations, and are useful in analyzing algebraic properties and deriving eigenvalue bounds for the corresponding preconditioned matrices. Numerical examples are given to show effectiveness of the banded Toeplitz preconditioners.     

Tangential frequency filtering decompositions for unsymmetric matrices

Summary. In this paper, tangential frequency filtering decompositions (TFFD) for unsymmetric matrices are introduced. Different algorithms for the construction of unsymmetric tangential frequency filtering decompositions are presented. These algorithms yield for a specified class of matrices equivalent decompositions. The convergence rates of an iterative scheme, which uses a sequence of TFFDs as preconditioners, are independent of the number of unknowns for this class of matrices. Several numerical experiments verify the efficiency of these methods for the solution of linear systems of equations which arise from the discretisation of convection-diffusion differential equations. Received April 1, 1996 / Revised version received July 4, 1996     

A review of infinite matrices and their applications

Infinite matrices, the forerunner and a main constituent of many branches of classical mathematics (infinite quadratic forms, integral equations, differential equations, etc.) and of the modern operator theory, is revisited to demonstrate its deep influence on the development of many branches of mathematics, classical and modern, replete with applications. This review does not claim to be exhaustive, but attempts to present research by the authors in a variety of applications. These include the theory of infinite and related finite matrices, such as sections or truncations and their relationship to the linear operator theory on separable and sequence spaces. Matrices considered here have special structures like diagonal dominance, tridiagonal, sign distributions, etc. and are frequently nonsingular. Moreover, diagonally dominant finite and infinite matrices occur largely in numerical solutions of elliptic partial differential equations.The main focus is the theoretical and computational aspects concerning infinite linear algebraic and differential systems, using techniques like conformal mapping, iterations, truncations etc. to derive estimates based solutions. Particular attention is paid to computable precise error estimates, and explicit lower and upper bounds. Topics include Bessel’s, Mathieu equations, viscous fluid flow, simply and doubly connected regions, digital dynamics, eigenvalues of the Laplacian, etc. Also presented are results in generalized inverses and semi-infinite linear programming.     

Solving mixed classical and fractional partial differential equations using short–memory principle and approximate inverses

The efficient numerical solution of the large linear systems of fractional differential equations is considered here. The key tool used is the short–memory principle. The latter ensures the decay of the entries of the inverse of the discretized operator, whose inverses are approximated here by a sequence of sparse matrices. On this ground, we propose to solve the underlying linear systems by these approximations or by iterative solvers using sequence of preconditioners based on the above mentioned inverses.     

Conical diffraction by multilayer gratings: a recursive integral equation approach

The paper is devoted to an integral equation algorithm for studying the scattering of plane waves by multilayer diffraction gratings under oblique incidence. The scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients in ?² coupled by special transmission conditions at the interfaces between different layers. Boundary integral methods lead to a system of singular integral equations, containing at least two equations for each interface. To deal with an arbitrary number of material layers we present the extension of a recursive procedure developed by Maystre for normal incidence, which transforms the problem to a sequence of equations with 2×2 operator matrices on each interface. Necessary and sufficient conditions for the applicability of the algorithm are derived.     

A brief historical introduction to matrices and their applications

This paper deals with the ancient origin of matrices, and the system of linear equations. Included are algebraic properties of matrices, determinants, linear transformations, and Cramer’s Rule for solving the system of algebraic equations. Special attention is given to some special matrices, including matrices in graph theory and electrical networks. It contains a wide variety of important materials accessible to college and even high school students and teachers at all levels.     

A method for least squares solution of systems with a cyclic rectangular coefficient matrix

An orthogonalization procedure is given for a sequence of vectors having the special feature that consecutive vectors are related by a_k+1 = Pa_k, (k = 1, 2, 3, …), where P is a unitary operator.This orthogonalization procedure is applied to the least squares solution of linear equations with a cyclic rectangular coefficient matrix. Furthermore, it is shown how the pseudoinverse of such matrices can be obtained.     

A CLASS OF NEW PARALLEL HYBRID ALGEBRAIC MULTILEVEL ITERATIONS

1. IntroductionConsider the large sparse system of linear equationsAx = b, (1.1)where, for a fixed positive integer cr, A e L(R") is a symmetric positive definite (SPD) matrir,having the bloCked formx,b E R" are the uDknwn and the known vectors, respectively, having the correspondingblocked formsni(ni S n, i = 1, 2,', a) are a given positthe integers, satisfying Z ni = n. This systemi= 1of linear equations often arises in sultable finite element discretizations of many secondorderseifad…     

A numerical method for solving partial differential algebraic equations

Linear systems of partial differential equations with constant coefficient matrices are considered. The matrices multiplying the derivatives of the sought vector function are assumed to be singular. The structure of solutions to such systems is examined. The numerical solution of initialboundary value problems for such equations by applying implicit difference schemes is discussed.     

On hybrid addition of matrices

The definition of the hybrid sum of arbitrary matrices is given, and it is shown that this definition generalizes the previous work done for the hybrid sum of Hermitian positive semidefinite matrices. It is shown that hybrid summability of two matrices is equivalent to the consistancy of set of linear equations. These equations are then used to derive many properties of the hybrid sum, in particular commutativity and associativity. The shorted operator and matrix gyration are generalized and their relationship to hybrid addition is discussed.     

Condition number of matrices derived from two classes of integral equations

We investigate some integral equations, i. a. the so-called Kupradze functional equations, where the two variables of the kernel belong to two different point sets. An extensive survey of the literature shows the various applications of these equations. By a discretization of the integral equations they are replaced by systems of linear algebraic equations. The condition number of the corresponding matrices is investigated, analytically and numerically. It is thereby quantitatively found in which way the condition of the matrices deteriorates when the two point sets are moved away from each other.     

Application of semiorthogonal spline wavelets and the Galerkin method to the numerical simulation of thin wire antennas

The Bubnov-Galerkin method based on spline wavelets is used to solve singular integral equations. For the resulting systems of linear algebraic equations, the properties of their coefficient matrices are examined. Sparse approximations of these matrices are constructed by applying a cutting barrier. The results are used to numerically analyze thin wire antennas. Numerical results are presented.     

A block-Lanczos method for large continuation problems

The computation of solution paths of large-scale continuation problems can be quite challenging because a large amount of computations have to be carried out in an interactive computing environment. The computations involve the solution of a sequence of large nonlinear problems, the detection of turning points and bifurcation points, as well as branch switching at bifurcation points. These tasks can be accomplished by computing the solution of a sequence of large linear systems of equations and by determining a few eigenvalues close to the origin, and associated eigenvectors, of the matrices of these systems. We describe an iterative method that simultaneously solves a linear system of equations and computes a few eigenpairs associated with eigenvalues of small magnitude of the matrix. The computation of the eigenvectors has the effect of preconditioning the linear system, and numerical examples show that the simultaneous computation of the solution and eigenpairs can be faster than only computing the solution. Our iterative method is based on the block-Lanczos algorithm and is applicable to continuation problems with symmetric Jacobian matrices. This revised version was published online in June 2006 with corrections to the Cover Date.     

A numerical method for solving boundary value problems for fractional differential equations

A numerical scheme, based on the Haar wavelet operational matrices of integration for solving linear two-point and multi-point boundary value problems for fractional differential equations is presented. The operational matrices are utilized to reduce the fractional differential equation to system of algebraic equations. Numerical examples are provided to demonstrate the accuracy and efficiency and simplicity of the method.     

Convergence and stability analyses for some vector extrapolation methods in the presence of defective iteration matrices

In two previous papers [10,11] convergence and stability results for the following vector extrapolation methods were presented: Minimal Polynomial Extrapolation, Reduced Rank Extrapolation, Modified Minimal Polynomial Extrapolation, and Topological Epsilon Algorithm. The analyses were carried out for vector sequences that include those arising from iterative methods for linear systems of equations having diagonalizable iteration matrices. In this paper the analyses of [10,11] are extended to vector sequences that include those arising from iterative methods for linear systems having defective iteration matrices. The results are illustrated with numerical examples. The analyses above naturally suggest some old and some new extensions of the well known power method, enabling one to obtain estimates for several dominant eigenvalues of a general matrix.     

Duality in conjugate gradient methods

In this paper we show that if the step (displacement) vectors generated by the preconditioned conjugate gradient algorithm are scaled appropriately they may be used to solve equations whose coefficient matrices are the preconditioning matrices of the original equations. The dual algorithms thus obtained are shown to be equivalent to the reverse algorithms of Hegedüs and are subsequently generalised to their block forms. It is finally shown how these may be used to construct dual (or reverse) algorithms for solving equations involving nonsymmetric matrices using only short recurrences, and reasons are suggested why some of these algorithms may be more numerically stable than their primal counterparts. This revised version was published online in June 2006 with corrections to the Cover Date.     

Subinverses of fuzzy matrices

A matrix operation is examined for fuzzy matrices and interesting properties of fuzzy matrices are obtained using the operation. Particularly some properties concerning subinverses and regularity of fuzzy matrices are given and the largest subinverse is shown by the properties. The properties are closely related to inverses of fuzzy matrices and fuzzy equations. Moreover fuzzy preorders are examined using the matrix operation and basic properties are obtained. The results are considered to be useful for the theory of fuzzy matrices.     

A new BCR method for coupled operator equations with submatrix constraint

In the present work, a new biconjugate residual algorithm (BCR) is proposed in order to compute the constraint solution of the coupled operator equations, in which the constraint solution include symmetric solution, reflective solution, centrosymmetric solution and anti-centrosymmetric solution as special cases. When the studied coupled operator equations are consistent, it is proved that constraint solution can be convergent to the exact solutions if giving any initial complex matrices or real matrices. In addition, when the studied coupled operator equations are not consistent, the least norm constraint solution above can also be computed by selecting any initial matrices. Finally, some numerical examples are provided for illustrating the effectiveness and superiority of new proposed method.