An Effective Java-based Algorithm to Compute Eigenvalues of Symmetric Toeplitz Matrices

Eigenvalues are a special set of scalars associated with a linear system of equations, that are sometimes also known as characteristic roots. In this work, a Java-based algorithm is developed to ﬁnd the eigenvalues for Toeplitz matrices using Adomian’s decomposition method (shortly, ADM). Illustrative examples will be examined to support the proposed analysis


INTRODUCTION
Many interesting phenomena in scientific and engineering applications are governed by algebraic equations. It is also known that there are two types of these equations, the linear, or nonlinear equations, and could be system. Most of these types of equations do not have an analytical solution, these equations should be solved by using numerical or approximate methods. In using numerical methods, like Sinc method, the solution reduces to the discrete system Ax = b, in which the coefficient matrix A is a combinations of Toeplitz matrices and diagonal matrices. Hence, it is a basic requirement to discuss the algebraic properties of these Topeplitz matrices. Toeplitz matrices arise in a variety of applications in Mathematics and Engineering. In particular, when the Sinc method is applied to discretize the differential equation (Ordinary, partial or integral), we can often obtain a linear system whose coefficients matrices are combinations of Toeplitz and diagonal matrices, see [1,2]. One of the main advantages of using the technique in this paper is discussing the stability of the discrete system being obtained by Sinc methodology, which is considered to be a basic requirement in finding bounds for the eigenvalues of Sinc matrices.
In the last decade, there has been some advanced developments including, Adomian decomposition method [3,4,5,6], Differential transform method [7], and Homotopy perturbation method [8] for solving various types algebraic equations. Eigenvalues are a special set of scalars associated with a linear system of equations, that are sometimes also known as characteristic roots. The determination of the eigenvalues of a system, like Ax = b, is important in physics and engineering, where it is equivalent to matrix diagonalization, and arises in such common applications as, stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems.
The basic motivation of this work, is to propose a new modification of the ADM to find eigenvalues for any n × n matrix. Therefore, as mentioned above, the objective of this paper is to find eigenvalues for Toeplitz matrices that comes from the theory of sinc function. At the beginning of the 80s, a new method, called ADM for solving various kinds of nonlinear equations had been proposed by Adomian [9,10]. The convergence of Adomian's method has been investigated by several authors [11]. ADM offers a reasonable, reliable solution to algebraic equations. The system is implemented for the full Java language, and is used to statically verify the correctness of Java. To demonstrate this we intend to solve two examples in the succeeding sections considering the symmetry of the given matrix. The outlines of the paper is as follows. In section 2, we derive the expression of the Toeplitz matrices from sinc function. The basic ideas of the ADM for solving algebraic equation is reviewed in Section 3. Numerical experiments are presented in section 4. Finally we give a concluding remark on the applications of the method.

TOEPLITZ MATRICES
We first give general expressions for the Toeplitz matrices associated with the Sinc discretization of various order. The goal of this section is to recall notations and definitions of the Sinc function that will be used in this paper. These are discussed in [1,2]. The Sinc function is defined on the whole real line IR by the k-th Sinc function is defined as The properties of Sinc functions have been extensively studied in [1,2]. The sinc method requires that the derivatives of sinc functions be evaluated at the nodes. Technical calculations provide the following results that will be useful in formulating the discrete system [1,2], and these quantities are delineated by δ In particular, the following convenient notation will be useful in formulating the discrete system jk ], q = 0, 1, 2, 3, ... i.e., the matrix whose jk− entry is given by δ kj and I (3) takes the form  which can be written in more general form as The eigenvalues of A are those numbers λ for which p λ (x) = 0. In general, from the fundamental theorem of algebra, there are n of these, in which our main goal in this paper is find these numbers. In most cases it is difficult to obtain an analytical solution of (2.6). Therefore the exploitation of numerical techniques for solving such equations becomes a main subject of considerable interests. Probably the most well-known and widely used algorithm to find a roots of equation (2.6) is Newton's method [5].

BASIC IDEA OF THE ADM
We apply the Adomian decomposition method (ADM) to find the smallest eigenvalue for a given matrix, for that we solve the obtained characteristic equation via the use of ADM [9,10]. To illustrate the basic idea of this method. where Aj are called the Adomian's polynomials obtained by the traditional formula where µ is the parameter introduced for convenience. Given a nonlinear function G(λ), the first few Adomian's polynomials are given by Wazwaz [3] developed a new algorithm for calculating Adomian polynomials for all forms of nonlinearity.
In this paper we are dealing with nonlinear polynomials only, so if G(λ) = λ 2 , then set λ = ∑ ∞ n=0 λn, so that Expanding the expression at the right hand side gives G(λ) = λ 2 0 + 2λ0λ1 + 2λ0λ2 + λ 2 1 + 2λ0λ3 + 2λ1λ2 + ... rearrange the above terms by grouping all terms with the sum of the subscript of the components of λn is the same, we obtain Expanding the expression at the right hand side, and rearrange terms, by grouping all terms with the sum of the subscript of the components of λn is the same, we obtain

Now back to the procedure, upon substituting equations (3.3) and (3.4) into equation (3.2) we arrive at
where An,j, An−1,j, ..., A2,j represent the Adomian polynomials for the nonlinear functions λ n , λ n−1 , ..., λ 2 respectively. To determine the components λn, n ≥, we first identify the zeroth component λ0 by all terms that are constant in equation (3.2). The remaining components of the series (3.3) can be determined in a way that each component is determined by using the preceding components, i.e., each term of the series (3.3) is given by by the following recursive relation λ0 = − an a n−1 λj+1 = − a 0 a n−1 An,j − a 1 a n−1 An−1,j − ... − a n−2 a n−1 A2,j, j = 0, 1, 2, ...

(3.7)
Finally, the solution λj can be approximated by the truncated series In computing λn, choosing large values for n, increasing the number of terms in the expression of Aj and this causes propagation of round off errors. The ADM reduces significantly the massive computation which may arise if discretization methods are used. The convergence series was investigated by several authors [12,11].

NUMERICAL EXAMPLES
In this section we shall illustrate the technique by different matrices. Dealing with matrices of known eigenvalues allow for more error analysis, in the first example we consider a 3 × 3 matrix in which we examine the accuracy and validity of our algorithm Using Mathematica, it is easy to verify that the smallest eigenvalue is | − 0.4351| = 0.4351, and the characteristic equation for finding the eigenvalues is given by To find the second eigenvalue, we treat a new characteristic function given by λ 4 + 13.1595λ 3 + 52.39λ 2 + 68.7453λ + 21.043 λ + 0.433766 then we repeat the same procedure as in above.