System Of Recurrence Relations Research Articles

Multipoint Super-Halley Type Approximation Algorithms in Banach Spaces

We use a new multipoint iteration of at least R-order three to solve a nonlinear operator equation in a Banach space. Under standard Newton-Kantorovich assumptions, we establish two Kantorovich convergence theorems using a new system of recurrence relations. We also give some explicit error bounds for this iteration.

A new type of recurrence relations for the secant method*

We apply the Secant method to solve nonlinear operator equations in Banach spaces. We establish a Newton-Kantorovich convergence theorem using a new system of recurrence relations and give an explicit expression for the a priori error bounds. Moreover, we apply our results to the numerical resolution of a nonlinear boundary value problem of second order and improve the error bounds obtained by other authors.

Remark on the convergence of the midpoint method under mild differentiability conditions

We establish a convergence theorem for the Midpoint method using a new system of recurrence relations. The purpose of this note is to relax its convergence conditions. We also given an example where our convergence theorem can be applied but other ones cannot.

Avoiding the computation of the second Fréchet-derivative in the convex acceleration of Newton's method

We introduce a new two-step method to approximate a solution of a nonlinear operator equation in a Banach space. An existence-uniqueness theorem and error estimates are provided for this iteration using Newton-Kantorovich-type assumptions and a technique based on a new system of recurrence relations. For a special choice of the parameter involved we use, our method is of fourth order.

Reduced Recurrence Relations for the Chebyshev Method

In this paper, we give sufficient conditions ensuring the convergence of the Chebyshev method in Banach spaces. We use a new system of recurrence relations which simplifies those given by Kantorovich for the Newton method or those given by Candela and Marquina for the Chebyshev and Halley methods.

A new family of multipoint methods of second order

We introduce a new family of multipoint methods of R-order two to approximate a solution of nonlinear operators equation in Banach spaces. An existence-uniqueness theorem and error estimates are provided for this family of iterations using Newton-Kantorovich-type assumptions and a technique based on a new system of recurrence relations. For a special choice of the parameter involved, we obtain the Midpoint method, whose order of convergence is three.

Solving two-point boundary value problems with spline functions

We consider a collocation method for the approximation of the solution of the nonlinear two-point boundary value problem y«(x) = f(x, y(x)), y(a) = A, y(b) = B, using splines of degree m≥3. The method which we shall use leads to a system of recurrence relations which can be solved by Newton's method. By obtaining asymptotic error bounds we verify a conjecture of Khalifa & Eilbeck, i.e. splines of even degree can give even better solutions than splines of odd degree in certain cases

The Problem of Phase Retrieval in Light and Electron Microscopy of Strong Objects: III. Developments of methods for numerical solution

In this paper we will develop procedures for solving the system of non-linear Volterra integral equations, which describe the relations between the complex wave functions in the exit pupil and the Fourier transforms of two intensity distributions in the image plane, corresponding to two images of the same object taken at different values of the defocusing of the optical system. The first method reduces the Volterra integral equations to a finite system of linear algebraic equations which can be solved successively. The second algorithm is based on the Newton-Kantorovich approximation for functional equations, which gives a system of recurrence relations by means of which the Volterra equations can be solved.

General problems of fast-electron transfer

A computation method is given in this article for the transfer of fast electrons in a plate of matter with nonuniform electric field. The kinetic equation with boundary conditions taken into account is transformed into a system of recurrence relations for the angular moments of the harmonic coefficients for the distribution function; this enables one to construct a simple computation algorithm. The Green's function is obtained for segments of a particle trajectory. Special features of calculations are given as well as results.

On certain expansions of the solutions of the general Lamé equation

1. In this paper I shall deal with the solutions of the Lamé equationwhen n and h are arbitrary complex or real parameters and k is any number in the complex plane cut along the real axis from 1 to ∞ and from −1 to −∞. Since the coefficients of (1) are periodic functions of am(x, k), we conclude ](5), § 19·4] that there is a solution of (1), y0(x), which has a trigonometric expansion of the formwhere θ is a certain constant, the characteristic exponent, which depends on h, k and n. Unless θ is an integer, y0(x) and y0(−x) are two distinct solutions of the Lamé equation.It is easy to obtain the system of recurrence relationsfor the coefficients cr. θ is determined, mod 1, by the condition that this system of recurrence relations should have a solution {cr} for whichk′ being the principal value of (1−k2)½