New modifications of Hansen–Patrick’s family with optimal fourth and eighth orders of convergence

Kansal, Munish; Kanwar, V.; Bhatia, Saurabh

doi:10.1016/j.amc.2015.07.101

Cited by 7 publications

(4 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Beberapa peneliti yang telah memodifikasi metode Hansen-Patrick dengan mereduksi turunan kedua menggunakan berbagai pendekatan, diantaranya: deret Taylor [9], tafsiran fungsi [10], dan tafsiran jumlah dua metode iterasi [11,12]. Selanjutnya, pada artikel ini dilakukan modifikasi metode iterasi Hansen-Patrick yang diberikan pada Persamaan (4).…”

Section: Pendahuluanunclassified

Modifikasi Metode Hansen-Patrick Dengan Orde Konvergensi Optimal Untuk Menyelesaikan Persamaan Nonlinier

Wartono

2022

KDMA

View full text Add to dashboard Cite

The Hansen-Patrick method is a third-order iterative method used to solve nonlinear equation. The method requires three evaluation of functions and has an efficiency index 31/3 » 1,4224. This study discusses a modification of the Hansen-Patrick method using the second order Taylor series. The second derivative is reduced using hyperbolic function with one parameter h. The aim of modification is to improve the convergence order of the Hansen-Patrick’s method. Based on the convergence analysis, the method has a fourth-order of convergence and envolve three evaluation of functions. So, its efficiency index is 41/3 » 1,5874. Numerical simulation is given to illustrate performance of the iterative method using six real functions. The performance of the iterative method include : a computational order of convergence, the number of iteration, evaluation of function, absolute error, and value of function, will be compared with Newton’s method, Halley’s method, Newton-Steffensen’s method, and Hansen-Patrick method. The numerical simulation shows that the performance of the method better than others

show abstract

Section: Pendahuluanunclassified

Modifikasi Metode Hansen-Patrick Dengan Orde Konvergensi Optimal Untuk Menyelesaikan Persamaan Nonlinier

Wartono

2022

KDMA

View full text Add to dashboard Cite

show abstract

“…In general, a common approach is to solve the equation y = f (x) iteratively for each value of y, using generalizations of the Newton-Raphson algorithm [4][5][6][7][8], or multipoint methods [9][10][11][12][13][14][15][16][17], or a combination of k-vector search with another inversion method, such as Newton-Raphson [18][19][20]. Moreover, special algorithms can be designed for a given function.…”

Section: Introductionmentioning

confidence: 99%

Fast Switch and Spline Function Inversion Algorithm with Multistep Optimization and k-Vector Search for Solving Kepler’s Equation in Celestial Mechanics

Tommasini

Olivieri

2020

Mathematics

View full text Add to dashboard Cite

Obtaining the inverse of a nonlinear monotonic function f(x) over a given interval is a common problem in pure and applied mathematics, the most famous example being Kepler’s description of orbital motion in the two-body approximation. In traditional numerical approaches, this problem is reduced to solving the nonlinear equation f(x)−y=0 in each point y of the co-domain. However, modern applications of orbital mechanics for Kepler’s equation, especially in many-body problems, require highly optimized numerical performance. Ongoing efforts continually attempt to improve such performance. Recently, we introduced a novel method for computing the inverse of a one-dimensional function, called the fast switch and spline inversion (FSSI) algorithm. It works by obtaining an accurate interpolation of the inverse function f−1(y) over an entire interval with a very small generation time. Here, we describe two significant improvements with respect to the performance of the original algorithm. First, the indices of the intervals for building the spline are obtained by k-vector search combined with bisection, thereby making the generation time even smaller. Second, in the case of Kepler’s equation, a multistep method for the optimized calculation of the breakpoints of the spline polynomial was designed and implemented in Cython. We demonstrate results that accurately solve Kepler’s equation for any value of the eccentricity e∈[0,1−ϵ], with ϵ=2.22×10−16, which is the limiting error in double precision. Even with modest current hardware, the CPU generation time for obtaining the solution with high accuracy in a large number of points of the co-domain can be kept to around a few nanoseconds per point.

show abstract

“…Notice that King's class of methods can be obtained from method (1.2), if δ = 0. Moreover, if δ = 1, we obtain methods proposed in [18] and [19]. From a computational point of view, the method (1.2) attains eighthorder of convergence using only four functional evaluations, viz.…”

Section: Introductionmentioning

confidence: 99%

“…It is worth noticing that method (1.2) is not changing when we use the conditions of Theorem 2.1 instead of stronger conditions used in [18]. Moreover, we can compute the computational order of convergence (COC) defined by…”

mentioning

confidence: 99%

On the Local Convergence of an Eighth-order Method for Solving Nonlinear Equations

Argyros

Kansal

Kanwar

2016

Annals of West University of Timisoara - Mathematics and Computer Science

Self Cite

View full text Add to dashboard Cite

Abstract. We present a local convergence analysis of an eighthorder method for approximating a locally unique solution of a nonlinear equation. Earlier studies such as have shown convergence of these methods under hypotheses up to the seventh derivative of the function although only the first derivative appears in the method. In this study, we expand the applicability of these methods using only hypotheses up to the first derivative of the function. This way the applicability of these methods is extended under weaker hypotheses. Moreover, the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

show abstract

New modifications of Hansen–Patrick’s family with optimal fourth and eighth orders of convergence

Cited by 7 publications

References 13 publications

Modifikasi Metode Hansen-Patrick Dengan Orde Konvergensi Optimal Untuk Menyelesaikan Persamaan Nonlinier

Modifikasi Metode Hansen-Patrick Dengan Orde Konvergensi Optimal Untuk Menyelesaikan Persamaan Nonlinier

Fast Switch and Spline Function Inversion Algorithm with Multistep Optimization and k-Vector Search for Solving Kepler’s Equation in Celestial Mechanics

On the Local Convergence of an Eighth-order Method for Solving Nonlinear Equations

Contact Info

Product

Resources

About