engleski

Augmentation of the generalized nxn eigenvalue equation to a generalized (n+1)x(n+1) eigenvalue equation

Generalised eigenvalue equation ( ) where and are hermitian matrices while is in addition positive definite is considered. This equation is augmented to a generalised eigenvalue equation ( ) where hermitian matrices and represent matrices and , respectively, augmented by one additional row and one additional column. It is shown how the eigenvalues and the eigenvectors of the augmented eigenvalue equation can be expressed in terms of the eigenvalues and the eigenvectors of the original eigenvalue equation. Operation count to obtain by this method all augmented eigenvalues and eigenvectors is of the order . Unless matrices involved are of some special kind such as sparse matrices or alike, this operation count is one order of magnitude smaller than operation count required by other presently known methods. In many practical cases operation count to obtain a single selected eigenvalue and/or eigenvector by this method is of the order . In the case of the generalised eigenvalue equation, all other methods usually require again operations, even if only a single eigenvalue and/or eigenvector is required. Thus in many cases of interest operation count to obtain a selected eigenvalue and/or eigenvector by this method is two orders of magnitude smaller than operation count required by other methods.

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