Mohamed A. Ramadan

On the matrix equation

Abstract In this paper we study iterative methods for finding the extremal positive definite solutions of the matrix equation X + A T X - 1 M A = I . First, a condition on the existence of a positive definite solution of this matrix equation is given. Then, the existence as well as the rate of convergence of some proposed algorithms to obtain the extremal positive definite solutions of this equation is presented. Moreover, a generalization of computationally simple and efficient known algorithm is applied for obtaining the extremal positive definite solutions. In addition, both the necessary and sufficient conditions for this matrix equation to have positive definite solution are presented. Numerical examples are given to illustrate the performance and the effectiveness of the algorithms.

Polynomial and nonpolynomial spline approaches to the numerical solution of second order boundary value problems

In this paper, quadratic and cubic polynomial and nonpolynomial spline functions based methods are presented to find approximate solutions to second order boundary value problems. Using these spline functions we drive a few consistency relations which to be used for computing approximations to the solution for second order boundary value problems. The present approaches have less computational cost. Convergence analysis of these methods is discussed. Two numerical examples are included to illustrate the practical usefulness of the proposed methods.

On the existence of a positive definite solution of the matrix equation

In this paper, an efficient and numerically stable algorithm for computing the positive definite solution of the nonlinear equation is proposed. Some properties of the solution are discussed as well as the sufficient conditions for the existence are obtained. Numerical examples are given to illustrate the accuracy of the proposed technique, which depend on iterative process

Numerical treatment for the modified burgers equation

In this paper, we consider the solution of the modified Burgers equation by using the collocation method with quintic splines. Applying the Von-Neumann stability analysis method we show that the proposed method is unconditionally stable. By conducting a comparison between the absolute error for our numerical results and the analytic solution of the modified Burgers equation we will test the accuracy of the proposed method.

A class of methods based on a septic non-polynomial spline function for the solution of sixth-order two-point boundary value problems

Two new second- and fourth-order methods based on a septic non-polynomial spline function for the numerical solution of sixth-order two-point boundary value problems are presented. The spline function is used to derive some consistency relations for computing approximations to the solution of this problem. The proposed approach gives better approximations than existing polynomial spline and finite difference methods up to order four and has a lower computational cost. Convergence analysis of these two methods is discussed. Three numerical examples are included to illustrate the practical use of our methods as well as their accuracy compared with existing spline function methods.

Partial eigenvalue assignment problem of high order control systems using orthogonality relations

In this paper, we present an explicit solution to the partial eigenvalue assignment problem of high order control system using orthogonality relations between eigenvectors of the matrix polynomial. Our solution can be implemented with only a partial knowledge of the spectrum and the corresponding left eigenvectors of the matrix polynomial. We show that the number of eigenvalues and eigenvectors that need to remain unchanged will not affected by feedback. A numerical example is given to illustrate the applicability and the practical usefulness of the proposed method.

Necessary and sufficient conditions for the existence of positive definite solutions of the matrix equation X+A T X −2 A=I

Necessary and sufficient conditions for the matrix equation X+A T X −2 A=I to have a real symmetric positive definite solution X are derived. Based on these conditions, some properties of the matrix A as well as relations between the solution X and A are derived.

Iterative positive definite solutions of the two nonlinear matrix equations X±A T X -2 A=I

In this paper iterative methods for obtaining positive definite solutions of the two matrix equations X+/-A^T X^-^2A=I are proposed. We show that under some conditions on the real square matrix A the constructed iterative methods converge to positive definite solutions for the two equations. Numerical examples are given to test and illustrate the performance of the algorithms in terms of convergence, accuracy as well as the efficiency.

On the explicit solutions of forms of the Sylvester and the Yakubovich matrix equations

In this paper, we consider the explicit solutions of two matrix equations, namely, the Yakubovich matrix equation V-AVF=BW and Sylvester matrix equations AV-EVF=BW,AV+BW=EVF and AV-VF=BW. For this purpose, we make use of Kronecker map and Sylvester sum as well as the concept of coefficients of characteristic polynomial of the matrix A. Some lemmas and theorems are stated and proved where explicit and parametric solutions are obtained. The proposed methods are illustrated by numerical examples. The results obtained show that the methods are very neat and efficient.

High order accuracy nonpolynomial spline solutions for 2μth order two point boundary value problems

Sixth order accurate method based on quintic nonpolynomial spline function for the numerical solution of 2μth order two point BVPs (where μ is a positive integer) is presented by transforming the problem into a system of μ second order two point BVPs. The nonpolynomial spline function is used to drive some consistency relations for computing approximation to the solution of this problem. The proposed approach gives better approximations than existing polynomial spline and finite difference methods and has a lower computational cost. Convergence analysis of the proposed method is discussed. Numerical examples are included to illustrate the practical usefulness of our method.