Strict localization of eigenvectors and eigenvalues

Determination of the eigenvalues and eigenvectors of a matrix is important in many areas of science, for example in web ranking [8, 15], computer graphics and visualization [24], quantum mechanics [5], statistics [12, 20], medicine, communications, construction vibration analysis [3, 23]. One of the most important numerical methods designed to calculate the eigenvalues and eigenvectors of matrix A is the power method [18, 21]. It is used to determine a maximum module eigenvalue of A and the corresponding eigenvector v. The limit of the product A w ‖Aw‖ , where w is a randomly chosen element, is the vector corresponding to the largest eigenvalue. The eigenvalue can be calculated from the Rayleigh quotient λ = 〈Av,v〉 〈v,v〉 . The most common method of solving the full eigenvalue problem, i.e. finding all the eigenvalues and corresponding eigenvectors, is the QR method [6, 13]. There are methods for locating eigenvalues such as Gerschgorin theorem [7] from 1931. This theorem allows to strictly locate the position of the eigenvalues of the matrix with real or complex coefficients. However, it does not allow to localize the eigenvectors. With the growing importance of these concepts there is a need to look for new methods of localizing simultaneously eigenvalues and eigenvectors. We do not know strict and efficient methods for locating eigenvectors of real or complex matrices, so we want to fill this gap. Our aim is to create and analyze a new method of strict location eigenvectors and eigenvalues with the use of interval arithmetic1. Using the fact that Corresponding author For language C++ one can use libraries such as ‘boost‘ [1] or ‘CAPD‘ [3].