Summary:
The problem of finding the eigenvalues and eigenvectors appears in many
practical applications of science. Especially, problems in physics and
engineering are typically modeled by systems of differential equations. Their
solution, in turn, is often expressed by means of eigenvalues and eigenvectors.
The symmetric generalized problem arises in various scientific fields, such as
geology, control theory, vibration analysis and image processing. The current
thesis is related to the standard and generalized eigenvalue problem and it is
divided into three main parts. The first part presents celebrated numerical
methods, whose usefulness is obvious in many mathematical issues. Efficient and
accurate methods for calculating eigenvalues and eigenvectors of random
matrices were unknown until the contrivance of the QR decomposition. There are
several methods for actually computing the QR decomposition, such as by means
of the Gram-Schmidt process, Householder transformations, or Givens rotations.
Each has a number of advantages and disadvantages. Special cases can be treated
efficiently with specialized algorithms. For instance, in the case of large
sparse matrices, the Lanczos algorithm allows one to compute eigenvalues and
eigenvectors in an effective manner.The second part consists of a thorough
presentation of numerical algorithms, targeting the standard eigenvalue
problem. Among the algorithms discussed in this section, the Power Method and
Inverse Iteration are in the spotlight if the largest eigenvalue and the
corresponding eigenvector are in need of computation, and the Implicit QR
Iteration, if all eigenvalues are required. The necessity of the QR
factorization is apparent at the current point.In the third part, the core
section of the paper, the generalized and symmetric generalized eigenvalue
problems are presented. The algorithms used previously take under consideration
the new input data. The well known QZ algorithm and the Arnoldi and Lanczos
Methods are discriminated because of their outstanding computational
functionality. Newer algorithms, like MDR and its diversified situations, are
also effective and convergence promising.
