Newton-type Iterative Methods And Geometric Iterative Algorithms For Solving Nonlinear Equations

Nonlinear problem play a great role in modern scientific computing disciplines. Many equations derived from practical problem are always the nonlinear forms. So how to solve these nonlinear problems appropriately becomes more and more hot in research disciplines. The main content of the article is constructing Newton-like iterative methods and geometric iterative algorithms for solving nonlinear equations1 Modification of Newton's method with higher-order convergence is presented. The modification of Newton's method is based on King's fourth-order method. The new method requires three-step per iteration. Analysis of convergence demonstrates that the order of convergence is 16. Some numerical examples illustrate that the algorithm is more efficient and performs better than classical Newton's method and other methods.2 This section presents a fifth-order iterative method as a new modification of Newton's method for finding multiple roots of nonlinear equations with unknown multiplicity m. Its convergence order is analyzed and proved. Moreover, several numerical examples demonstrate that the proposed iterative method is superior to the existing methods.3 Based on the fourth-order method of Li et al. [Li, Zheng, Zhao, a variant of Steffenson's method of fourth-order convergence and its applications, Appl. Math. Comput. 216(2010), 1978-1983], a robust seventh-order three-step method which is free from derivative and reaches the efficiency index 1.626 is presented.4 A class of three-step eighth-order root solvers is constructed in this section. Our aim is fulfilled by using an interpolatory rational function in the third step of a three-step cycle. Each method of the class reaches the optimal efficiency index according to the Kung-Traub Conjecture concerning multi-point iterative methods without memory. Moreover, the class is free from derivative calculation per full iteration which is of grave importance in engineering problems. One method of the class is established analytically. To test its accuracy, we apply it to a lot of nonlinear equations. Numerical examples suggest that the novel class of derivative-free methods is much faster than the existing methods in literature.5 In [Petkovic, SIAM J.NUMER. ANAL. 47(2010), pp.4402-4414], a general class of n point iterative methods for solving nonlinear equations is constructed. The author proved that these methods have the convergence order 2n , requiring only n+ 1 function evaluations per iteration. In this note, we show that the general class of n point iterative methods given by the author do not support the Kung-Traub conjecture(1974) on the upper bound 2n of the order of multipoint methods based on n+ 1 function evaluations.6 On point projection and inversion problem, Hu et al. have presented an algorithm for orthogonal projection onto curves and surfaces in their paper (see Computer Aided Geometric Design, 22(2005)251-260). In this section, on point projection problem, we have formally proved that projecting a point onto a planar parameter curve is second-order globally convergent. On point inversion problem, we have formally proved that the inversion problem is third-order globally convergent. A new method is presented for computing the minimum distance between a point and a spatial parametric curve. It consists of a geometric iteration which converges faster than the existing Newton method, and whose sensitivity to the choice of initial values is nonexistence. We have proved that projecting a point onto a spatial parametric curve is second-order globally convergent. A new method is presented for computing the minimum distance between a point and a spatial parametric surface. It consists of a geometric iteration which converges faster than the existing Newton method, and whose sensitivity to the choice of initial values is nonexistence. We can know that projecting a point onto a spatial parametric surface is second-order globally convergent.