The parameter functionalization method in the eigenvalue problem (Q1594329)

The first part of the article deals with iterative algorithms of solving the eigenvalue problem \(Ax=\lambda x\) with a complete continuous linear operator \(A\) in a Banach space \(E\). Under the assumption that \(A\) has a nonzero simple (real or complex) eigenvalue \(\lambda_*\) and \(X_*\) is the corresponding eigenvector the authors reduce the problem to the operator equation \(Ax-f(x) x=0\) where \(f\) is a suitable continuous linear functional (this pass is called method of parameter functionalization). Then the basic and modified Newton-Kantorovich iterations for this operator equation are considered. In the second part the problem \(A(\varepsilon) x=\lambda x\) is considered. In this case iterations discussed in the first part depend on the parameter \(\varepsilon\) and are asymptotic approximations to the exact solution. The order of these approximations is easily estimated in dependence on \(n\).