Polynomials Root-Finding Using a SLEFE-Based Clipping Method

For finding the real roots of a polynomial,we propose a clipping algorithmcalled SLEFEclipping and an isolation algorithmcalled SLEFEisolation algorithm.Ateach iterative step,the SLEFEclipping algorithm generates two broken lines boundingthe given polynomial.Then,a sequence of intervals can be obtained by computing theintersection of the sequence of broken lines with the abscissa axis.The sequence ofthese intervals converges to the root with a convergence rate of 2.Numerical examplesshow that SLEFE clipping requires fewer iterations and less computation time thancurrent algorithms,and the SLEFE isolation algorithm can compute all intervals thatcontain the roots rapidly and accurately.

A Fourth-order Covergence Newton-type Method

A fourth-order convergence method of solving roots for nonlinear equation, which is a variant of Newton＇s method given. Its convergence properties is proved. It is at least fourth-order convergence near simple roots and one order convergence near multiple roots. In the end, numerical tests are given and compared with other known Newton and Newton-type methods. The results show that the proposed method has some more advantages than others. It enriches the methods to find the roots of non-linear equations and it is important in both theory and application.

Improved Ostrowski-Like Methods Based on Cubic Curve Interpolation

In this paper, we derive two higher order multipoint methods for solving nonlinear equations. The methodology is based on Ostrowski’s method and further developed by using cubic interpolation process. The adaptation of this strategy increases the order of Ostrowski’s method from four to eight and its efficiency index from 1.587 to 1.682. The methods are compared with closest competitors in a series of numerical examples. Moreover, theoretical order of convergence is verified on the examples.

Solving Systems of Transcendental Equations Involving the Heun Functions

The Heun functions have wide application in modern physics and are expected to succeed the hypergeometrical functions in the physical problems of the 21st century. The numerical work with those functions, however, is complicated and requires filling the gaps in the theory of the Heun functions and also, creating new algorithms able to work with them efficiently. We propose a new algorithm for solving a system of two nonlinear transcendental equations with two complex variables based on the Müller algorithm. The new algorithm is particularly useful in systems featuring the Heun functions and for them, the new algorithm gives distinctly better results than Newton’s and Broyden’s methods. As an example for its application in physics, the new algorithm was used to find the quasi-normal modes (QNM) of Schwarzschild black hole described by the Regge-Wheeler equation. The numerical results obtained by our method are compared with the already published QNM frequencies and are found to coincide to a great extent with them. Also discussed are the QNM of the Kerr black hole, described by the Teukolsky Master equation.

New Fourth Order Iterative Methods Second Derivative Free

In a recent paper, Noor and Khan [M. Aslam Noor, & W. A. Khan, (2012) New Iterative Methods for Solving Nonlinear Equation by Using Homotopy Perturbation Method, Applied Mathematics and Computation, 219, 3565-3574], suggested a fourth-order method for solving nonlinear equations. Per iteration in this method requires two evaluations of the function and two of its first derivatives;therefore, the efficiency index is 1.41421 as Newton’s method. In this paper, we modified this method and obtained a family of iterative methods for appropriate and suitable choice of the parameter. It should be noted that per iteration for the new methods requires two evaluations of the function and one evaluation of its first derivatives, so its efficiency index equals to 1.5874. Analysis of convergence shows that the methods are fourth-order. Several numerical examples are given to illustrate the performance of the presented methods.

Ray-triangular Bezier patch intersection using hybrid clipping algorithm

In this paper, we present a novel geometric method for efficiently and robustly computing intersections between a ray and a triangular Bezier patch defined over a triangular domain, called the hybrid clipping （HC） algorithm. If the ray pierces the patch only once, we locate the parametric value of the intersection to a smaller triangular domain, which is determined by pairs of lines and quadratic curves, by using a multi-degree reduction method. The triangular domain is iteratively clipped into a smaller one by combining a subdivision method, until the domain size reaches a prespecified threshold. When the ray intersects the patch more than once, Descartes＇ rule of signs and a split step are required to isolate the intersection points. The algorithm can be proven to clip the triangular domain with a cubic convergence rate after an appropriate preprocessing procedure. The proposed algorithm has many attractive properties, such as the absence of an initial guess and insensitivity to small changes in coefficients of the original problem. Experiments have been conducted to illustrate the efficacy of our method in solving ray-triangular Bezier patch intersection problems.

A High-Order Newton-Like Method

This paper gives a new iterative method to solve the non-linear equation. We prove that this method has the asymptotic convergent order. When the iterative times exceed 2,only one evaluation of the function and one of its first derivative is required by each iteration of the method.Therefore the new method is better than Newton＇s method.