On the numerical radius of matrices and its application to iterative solution methods

Uses of the numerical radius in the analysis of basic iterative solution methods, of the SOR method for quasi-Hermitian positive definite matrices (not being consistently ordered) and of maximal eigenvalues of symmetric positive definite matrices using incomplete block-matrix factorizations are presented.     

On the numerical radius and its applications

We give a brief account of the numerical radius of a linear bounded operator on a Hilbert space and some of its better-known properties. Both finite- and infinite- dimensional aspects are discussed, as well as applications to stability theory of finite-difference approximations for hyperbolic initial-value problems.     

The numerical radius and specttural matrices

In this paper we investigate spectral matrices, i.e., matrices with equal spectral and numerical radii. Various characterizations and properties of these matrices are given.     

The numerical radius and specttural matrices

In this paper we investigate spectral matrices, i.e., matrices with equal spectral and numerical radii. Various characterizations and properties of these matrices are given.     

Direct and iterative methods for the numerical solution of mixed integral equations

In this paper we analyze and compare two classical methods to solve Volterra–Fredholm integral equations. The first is a collocation method; the second one is a fixed point method. Both of them are proposed on a particular class of approximating functions. Precisely the first method is based on a linear spline class approximation and the second one on Schauder linear basis. We analyze some problems of convergence and we propose some remarks about the peculiarities and adaptability of both methods. Numerical results complete the work.     

On the solution set of convex problems and its numerical application

The solution set of a convex problem is enough interested. In this paper we have a discussion to determine it. The conditions and proofs in this note essentially differs from the previous well-known works. We also have a new result determining the solution of minimum norm for quadratic problems.     

On the Vykhandu-Levin iterative method for numerical solution of nonlinear systems

An iterative method for the solution of systems of nonlinear equations initiated by Vykhandu and investigated by Levin is discussed. It is shown that there is a flaw in the proof of Levin that the method is third-order convergent. Moreover, it is proved that the correct order of the method is only two.     

Linear operators preserving the higher numerical radius of matrices

A linear operator T on a matrix space is said to be unital if T(I) = I. In this note we characterize the unital lineart operators on matrix spaces that preserve the k-numerical radius. Using the results obtained we easily determine the structure of all linear operators on the space of n × n complex matrices that preserve the k-numerical range. This completes the work of Pierce and Watking, who obtained the results for the cases when n ≠ n2k.     

Equivalent iterative methods for p-cyclic matrices

An overview is given of the simplifications which arise when p-cyclic systems are solved by iterative methods. Besides the classic iterative methods, we treat the Chebyshev semi-iterative method and the OR and MR variants of the class of Krylov subspace methods. Particular emphasis is given to equivalent iterations applied to the cyclically reduced system.     

On the numerical solution of Korteweg-de Vries equation by the iterative splitting method

In this paper, we apply the method of iterative operator splitting on the Korteweg-de Vries (KdV) equation. The method is based on first, splitting the complex problem into simpler sub-problems. Then each sub-equation is combined with iterative schemes and solved with suitable integrators. Von Neumann analysis is performed to achieve stability criteria for the proposed method applied to the KdV equation. The numerical results obtained by iterative splitting method for various initial conditions are compared with the exact solutions. It is seen that they are in a good agreement with each other.     

On the application of numerical methods to the solution of nonlinear second-order differential equations with random deviations of argument

We consider the application of the Krylov-Bogolyubov-Mitropol’skii asymptotic method and Runge-Kutta methods to the investigation of oscillating solutions of quasilinear second-order differential equations with random deviations of argument. For specific equations, we obtain approximate numerical solutions and characteristics of random oscillations. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10, pp. 1433–1441, October, 1999.     

Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations

Summary We discuss block matrices of the formA=[A _ij ], whereA _ij is ak×k symmetric matrix,A _ij is positive definite andA _ij is negative semidefinite. These matrices are natural block-generalizations of Z-matrices and M-matrices. Matrices of this type arise in the numerical solution of Euler equations in fluid flow computations. We discuss properties of these matrices, in particular we prove convergence of block iterative methods for linear systems with such system matrices.     

On the iterative solution of the algebraic equations in fully implicit Runge-Kutta methods

This paper deals with the iterative solution of stage equations which arise when some fully implicit Runge-Kutta methods, in particular those based on Gauss, Radau and Lobatto points, are applied to stiff ordinary differential equations. The error behaviour in the iterates generated by Newton-type and, particularly, by single-Newton schemes which are proposed for the solution of stage equations is studied. We consider stiff systems y'(t) = f(t,y(t)) which are dissipative with respect to a scalar product and satisfy a condition on the relative variation of the Jacobian of f(t,y) with respect to y, similar to the condition considered by van Dorsselaer and Spijker in [7] and [17]. We prove new convergence results for the single-Newton iteration and derive estimates of the iteration error that are independent of the stiffness. Finally, some numerical experiments which confirm the theoretical results are presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

Approximation of kernel matrices by circulant matrices and its application in kernel selection methods

This paper focuses on developing fast numerical algorithms for selection of a kernel optimal for a given training data set. The optimal kernel is obtained by minimizing a cost functional over a prescribed set of kernels. The cost functional is defined in terms of a positive semi-definite matrix determined completely by a given kernel and the given sampled input data. Fast computational algorithms are developed by approximating the positive semi-definite matrix by a related circulant matrix so that the fast Fourier transform can apply to achieve a linear or quasi-linear computational complexity for finding the optimal kernel. We establish convergence of the approximation method. Numerical examples are presented to demonstrate the approximation accuracy and computational efficiency of the proposed methods.     

On the determinant of quaternionic polynomial matrices and its application to system stability

In this paper, we propose a definition of determinant for quaternionic polynomial matrices inspired by the well‐known Dieudonné determinant for the constant case. This notion allows to characterize the stability of linear dynamical systems with quaternionic coefficients, yielding results which generalize the ones obtained for the real and complex cases. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimal successive overrelaxation iterative methods for P-cyclic matrices

Summary We consider linear systems whose associated block Jacobi matricesJ _p are weakly cyclic of indexp. In a recent paper, Pierce, Hadjidimos and Plemmons [13] proved that the block two-cyclic successive overrelaxation (SOR) iterative method is numerically more effective than the blockq-cyclic SOR-method, 2<qp, if the eigenvalues ofJ _p ^p are either all non-negative or all non-positive. Based on the theory of stationaryp-step methods, we give an alternative proof of their theorem. We further determine the optimal relaxation parameter of thep-cyclic SOR method under the assumption that the eigenvalues ofJ _p ^p are contained in a real interval, thereby extending results due to Young [19] (for the casep=2) and Varga [15] (forp>2). Finally, as a counterpart to the result of Pierce, Hadjidimos and Plemmons, we show that, under this more general assumption, the two-cyclic SOR method is not always superior to theq-cyclic SOR method, 2<qp.Dedicated to R. S. Varga on the occasion of his 60th birthdayResearch supported in part by the Deutsche Forschungsgemeinschaft     

New iterative methods for solution of the eigenproblem

Summary Recently developed minimization methods are applied to the minimization of the Rayleigh quotient of a general eigensystem, producing efficient iterative methods for finding the extremal eigenvalues and associated eigenvectors. Numerical results are quoted for systems of widely differing order, and an assessment of the methods is made, together with how they compare against other well-known methods.Rolls Royce Ltd., Derby, formerly University of Leeds.University of Leeds.     

On the solvability and numerical methods for solution of linear integro-algebraic equations

We study solvability conditions for a system of Volterra equations with some identically degenerate or rectangular matrix at the main term. Connection is discussed of the solvability conditions and applicability of numerical methods for solving these systems. In particular, the conditions of the convergence of the least squares method with the error functional defined in Sobolev spaces are presented.