In this paper, we report the application of two parameter coupled alternating group explicit (CAGE) iteration and Newton-CAGE iteration methods for the cubic spline solution of non-linear differential equation u" = f(r,u,u') subject to given natural boundary conditions. The error analysis for CAGE iteration method is discussed in details. We compared the results of proposed CAGE iteration metho...

The problem of recovering a parameter function based on measurements of solutions of a system of partial diierential equations in several space variables leads to a number of computational challenges. Upon discretization of a regularized formulation a large, sparse constrained optimization problem is obtained. Typically in the literature , the constraints are eliminated and the resulting uncons...

In this paper a second-order method for solving large-scale strongly convex `1-regularized problems is developed. The proposed method is a NewtonCG (Conjugate Gradients) algorithm with backtracking line-search embedded in a doubly-continuation scheme. Worst-case iteration complexity of the proposed Newton-CG is established. Based on the analysis of Newton-CG, worstcase iteration complexity of t...

By making use of duality mappings, we formulate an inexact Newton– Landweber iteration method for solving nonlinear inverse problems in Banach spaces. The method consists of two components: an outer Newton iteration and an inner scheme providing the increments by applying the Landweber iteration in Banach spaces to the local linearized equations. It has the advantage of reducing computational w...

Newton’s iteration has quadratic convergence for simple roots. We present a Newton-based iteration scheme with quadratic convergence for multiple roots of systems of analytic functions. This is a report on work in progress. 1. Newton Iteration, Approximate Roots and γ-Theorems 1.1. Newton Iteration. Let f : Cn → Cn be an analytic function. Newton’s method for solving f = 0 consists in approxima...

‎Newton method is one of the most famous numerical methods among the line search‎ ‎methods to minimize functions. ‎It is well known that the search direction and step length play important roles ‎in this class of methods to solve optimization problems. ‎In this investigation‎, ‎a new modification of the Newton method to solve ‎unconstrained optimization problems is presented‎. ‎The significant ...

An iterative formula based on Newton’s method alone is presented for the iterative solutions of equations that ensures convergence in cases where the traditional NewtonMethodmay fail to converge to the desired root. In addition, themethod has super-quadratic convergence of order 2.414 (i.e., 1 + √2). Newton method is said to fail in certain cases leading to oscillation, divergence to increasing...

Abstract. In this paper, we develop a concrete algorithm for phase retrieval, which we refer to as GaussNewton algorithm. In short, this algorithm starts with a good initial estimation, which is obtained by a modified spectral method, and then update the iteration point by a Gauss-Newton iteration step. We prove that a re-sampled version of this algorithm quadratically converges to the solution...

The 1669-1670 Newton-Raphson’s method is still used to solve equations systems and unconstrained optimization problems. Since this method, some other algorithms inspired by Newton’s have been proposed: in 1839 Chebyshev developped a high order cubical convergence algorithm, and in 1967 Shamanskii proposed an acceleration of Newton’s method. By considering a Newton-type methods as displacement d...

We study multihomogeneous analytic functions and a multihomogeneous Newton’s method for finding their zeros. We give a convergence result for this iteration and we study two examples: the evaluation map and the generalized eigenvalue problem.