Iterative Methods for Approximating Eigenvalues and Eigenvectors

In this book some numerically iterative methods for approximating the eigenvalues and eigenvectors of the matrices would be studied. The methods such as power and QR method were studied. The power method approximates dominant eigenvalues (in magnitude) and corresponding eigenvectors. The inverse power method approximates the smallest eigenvalues (in magnitude) as well as corresponding eigenvectors and the shifted inverse power method approximates the intermediate means neither dominant nor smallest eigenvalues. The QR method approximates all eigenvalues, but inverse power method was used to approximate the corresponding eigenvectors. In approximating the eigenvalues of the matrix it is better to Transform the matrix by using similarity transformations to Hessenberg if the matrix was symmetric and to Tridiagonal if the is nonsymmetric. Transforming the matrix of large size (n4) were very difficult to approximate manually, therefore MATLAB would be used to approximate the eigenvalue and corresponding eigenvector. If the magnitude of two approximated eigenvalues were very close to each other the convergence were very slow,so the technique called shifting was applied to the matrix.