The matrix eigenvalue problem plays an important role in engineering applications. In vibration analysis, for example, eigenvalues are directly related to the system’s natural frequencies, while the eigenvectors represent the mode shapes. Eigenvalues play an equally significant role in numerical methods. For example, in the iterative solution of linear systems via Jacobi and Gauss–Seidel methods (Chapter 4), the eigenvalues of the Jacobi iteration matrix, or the Gauss–Seidel iteration matrix, not only determine whether or not the respective iteration will converge to a solution, but they also establish the rate of convergence of the method. In this chapter, we will present numerical methods to approximate the ...
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