The Ulm-like Method For Nonlinear Equations And Inverse Eigenvalue Problems

With the rapid development of science and technology and the widespread popularity of computer application, the method for solving the nonlinear equations is widely used in economy, computer science, information science, physics, the field of life science and so on. This paper mainly studies the solution of general nonlinear operator equations and the numerical method for solving the inverse eigenvalue problems that can be transformed into nonlinear equations problems.Chapter 1 introduces the development of general nonlinear operator equations and inverse eigenvalue problems, the relevant preliminary knowledge, includs related concepts such as inverse eigenvalue problems, order of convergence, convergence condition and the related conclusion in Banach space. Using the majorizing sequence to prove semilocal convergence and two common methods to construct the majorizing sequence are also presented. The paper structure is shown at last.In Chapter 2, an Ulm-like method is proposed for solving general nonlinear operator equations. This method avoids computing Jacobian matrices and avoids solving Jacobian equations. Under some conditions, we prove that the sequence produced by this Ulm-like method converges locally to the solution of the equations.Chapter 3 studies the semi-local convergence problem of the Newton-like method for solving the inverse eigenvalue problems. We try to use the majorizing sequence to study the convergence issue. Assuming the distinction of the given eigenvalues and the nonsingularity of the Jacobian matrix at the initial point, a Kantorovich-type convergence criterion based on the information around the initial point is established. Comparing with other known convergence results of the numerical methods for solving the inverse eigenvalue problems, our convergence results out of the dependence of the solution of inverse eigenvalue problems.