Eigenvalues of tridiagonal matrix using Strum Sequence and Gerschgorin theorem

In this paper, computational efficient technique is proposed to calculate the eigenvalues of a tridiagonal system matrix using Strum sequence and Gerschgorin theorem. The proposed technique is applicable in various control system and computer engineering applications. KeywordsEigenvalues, tridiagonal matrix, Strum sequence and Gerschgorin theorem. I.INTRODUCTION Solving tridiagonal linear systems is one of the most important problems in scientific computing. It is involved in the solution of differential equations and in various areas of science and engineering applications such control system [1] and computer science [2, 3]. It is also occur in a wide variety of applications, such as the construction of certain splines and the solution of boundary value problems. There are various numerical techniques available in the literature, which are useful for determining eigenvalues of real symmetric matrices [4]. In most of these methods, given system matrix is converted into tridiagonal form. There are various methods also given in the literature for determining eigenvalues of a tridiagonal matrix [4]. In this method, strum sequence and bisection method is used to determine the eigenvalues of a given real symmetric triangular matrix. It is observed that it require large number of iterations to compute the eigenvalues. It is observed that these iterations can be reduced by using well known Gerschgorin theorem [4]. In [5], technique is presented to identify the eigenvalues on the right half the s-plane using Gerschgorin theorem [4]. The extension to this approach is given in [6]. One of the leading methods for computing the eigenvalues of a real symmetric matrix is Given’s method [4]. In that method, after transforming the matrix into diagonal form say, ' ' S , the leading principal minors of S I λ − form a strum sequence. Then, using bisection approach, change of sign in various strum sequence is observed. Further, based on this, eigenvalue can be determined by repeatedly using bisection method. In [7], various applications are presented based on Gerschgorin theorem. Here, in this paper, similar to these existing applications, we have used Gerschgorin theorem in Strum sequence to determine eigenvalues in computationally efficient manner. In order to show the comparative result, we have considered the example which was earlier considered in [4]. II. Givens Method for Symmetric Matrices [4]: Let A be a real, symmetric matrix. The Givens method uses the following steps: (a) Reduce A to a tridiagonal form using plane rotations (b) Form a strum sequence, study the changes in sign in the sequences and find the eigenvalues. The reduction to a tridiagonal form is achieved by using the orthogonal transformation. Suppose the orthogonal matrix is given as T.D.Roopamala et al. / International Journal on Computer Science and Engineering (IJCSE) ISSN : 0975-3397 Vol. 3 No. 12 December 2011 3722