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Iterative Methods for Computing Eigen values and Eigen vectors with an Improved

Farnosh Izadi
Associate Professor, Islamic Azad University, Sowmesara Branch, Sowmesara, Iran
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One of the oldest techniques for solving eigenvalue problems is the so-colled power method. In this study, we examine some numerical iterative methods for computing the eigenvalues and eigenvectors of real matrices. The two methods examined here range from the simple power iteration method, and we produced an improvement in the convergence of them. Our work is based on choosing of initial vector in iterative methods for acceleration purpose. Finally, some examples are presented to illustrate the method and results discussed.


Aitken Acceleration, Power method, Rayleigh method, Dominant eigen value


Eigenvalues and eigenvectors play an important part in the applications of linear algebra. The naive method of finding the eigenvalues of a matrix involves finding the roots of the characteristic polynomial of the matrix. In industrial sized matrices, however, this method is not feasible, and the eigenvalues must be obtained by other means. Fortunately, there exist several other techniques for finding eigenvalues and eigenvectors of a matrix, some of which fall under the realm of iterative methods. These methods work by repeatedly refining approximations to the eigenvectors or eigenvalues, and can be terminated whenever the approximations reach a suitable degree of accuracy. Iterative methods form the basis of much of modern day eigenvalue computation. In this paper, we outline two such iterative methods, and summarize their derivations, procedures, and advantages. The methods to be examined are the power iteration method and the Rayleigh quotient method.
This paper is meant to be an improvement of existing algorithms for the eigenvalue computation problem. Section 2 of this paper provides a brief review of some of the linear algebra background required and algorithms of two existing methods. In Section 3, we have presented us idea on the existing algorithms. In Section 4, have been produced 4 numerical examples. Finally, in Section 5, we summarized some remarks and mention some of the additional algorithm refinements that are used in practice. We restrict our attention to real-valued, square matrices with a full set of real eigenvalues.






In this section, we consider three examples that we used eps =10-4 for all of them. All results have been computed using MATLAB [9].
Exp4.1 Let the following matrix of A with the largest eigenvalue of λ=3.


Here, we used the new initial vector for the two existing methods for the approximation of dominant eigenvalue. Then, we have used Aitken method for acceleration purpose. Numerical examples show the new method is faster.


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