An overview on the eigenvalue computation for matrices

This paper sketches the research developments in the area of computational methods for solving the eigenvalue problems and how the methods developed relate to each other as they evolve over centuries. This is an attempt to write a complete overview on the research on computational aspects of eigenvalue problem, emphasize the history of methods that still play a role and some of those that no longer are considered to be on the main track but are somehow related to the present techniques in some smaller steps. This contribution brings out the state-of-the-art of the algorithms for solving large-scale eigenvalue problems for both symmetric and nonsymmetric matrices separately, thereby clearly drawing a comparison between the differences in the algorithms in practical use for the two. Some of the methods or modifications to the earlier methods that have been notable at the turn of the 21st century will also be covered through this paper under "Modem Approaches". Also as the standard eigenvalue problem has become better understood, in a numerical sense, progress has been made in the generalized eigenvalue problem and this will briefly cover developments in solving these problems too.