On the calculation of eigenvalues by the method of steepest descent
Abstract
L. V. Kantorovich proposed a new direct method for solving problems of the calculus of variations, which he called the steepest descent method. As L. V. Kantorovich showed, the method can be successfully used for the approximate solution of linear functional equations in a Hilbert space. In addition, L. V. Kantorovich applied the steepest descent method to the problem of eigenvalues of completely continuous operators. In application to this problem, the method consists of the following. In this note, a proof of the convergence of the process is given without the assumption of closeness of x0 to x*. In addition, a comparison of the steepest descent method with the known iterative method for calculating eigenvalues is made, and a multi-step version of the method under consideration is investigated. In what follows, the assumption that m = 0 is not used.
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