On the Derivation of Higher Order Root-Finding Methods

2007 ◽  
Author(s):  
Mohammed A. Hasan
Keyword(s):  
Higher Order ◽  
Root Finding ◽  
Order Root

scholarly journals Higher-Order Root-Finding Algorithms and Their Basins of Attraction

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Amir Naseem ◽  
M. A. Rehman ◽  
Thabet Abdeljawad
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Complex Polynomials ◽  
Convergence Criteria ◽  
Root Finding ◽  
Variational Iteration ◽  
Order Root ◽  
Iteration Technique

In this paper, we proposed and analyzed three new root-finding algorithms for solving nonlinear equations in one variable. We derive these algorithms with the help of variational iteration technique. We discuss the convergence criteria of these newly developed algorithms. The dominance of the proposed algorithms is illustrated by solving several test examples and comparing them with other well-known existing iterative methods in the literature. In the end, we present the basins of attraction using some complex polynomials of different degrees to observe the fractal behavior and dynamical aspects of the proposed algorithms.

Changes in Higher Order Aberrations after Wavefront-Guided Prk for Correction of Low to Moderate Myopia and Myopic Astigmatism: Two-Year Follow-Up

2007 ◽  
Vol 17 (4) ◽  
pp. 507-514 ◽  
Author(s):  
D. Wygledowska-Promienska ◽  
I. Zawojska
Keyword(s):  
Visual Acuity ◽  
Excimer Laser ◽  
Spherical Aberration ◽  
Higher Order ◽  
Control Group ◽  
Higher Order Aberrations ◽  
Order Root ◽  
Group 2 ◽  
Myopic Astigmatism ◽  
Group 1

Purpose To assess efficacy, safety, and changes in higher order aberrations after wavefront-guided photorefractive keratectomy (PRK) in comparison with conventional PRK for low to moderate myopia with myopic astigmatism using a WASCA Workstation with the MEL 70 G-Scan excimer laser. Methods A total of 126 myopic or myopic-astigmatic eyes of 112 patients were included in this retrospective study. Patients were divided into two groups: Group 1, the study group; and Group 2, the control group. Group 1 consisted of 78 eyes treated with wavefront-guided PRK. Group 2 consisted of 48 eyes treated with spherocylindrical conventional PRK. Results Two years postoperatively, in Group 1, 5% of eyes achieved an uncorrected visual acuity (UCVA) of 0.05; 69% achieved a UCVA of 0.00; 18% of eyes experienced enhanced visual acuity of −0.18 and 8% of −0.30. In Group 2, 8% of eyes achieved a UCVA of 0.1; 25% achieved a UCVA of 0.05; and 67% achieved a UCVA of 0.00 according to logMAR calculation method. Total higher-order root-mean square increased by a factor 1.18 for Group 1 and 1.6 for Group 2. There was a significant increase of coma by a factor 1.74 in Group 2 and spherical aberration by a factor 2.09 in Group 1 and 3.56 in Group 2. Conclusions The data support the safety and effectiveness of the wavefront-guided PRK using a WASCA Workstation for correction of low to moderate refractive errors. This method reduced the number of higher order aberrations induced by excimer laser surgery and improved uncorrected and spectacle-corrected visual acuity when compared to conventional PRK.

scholarly journals Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

2015 ◽  
Vol 2015 ◽  
pp. 1-19 ◽  
Author(s):  
Krzysztof Gdawiec ◽  
Wiesław Kotarski ◽  
Agnieszka Lisowska
Keyword(s):  
Iteration Process ◽  
Higher Order ◽  
Point Of View ◽  
Complex Polynomials ◽  
Root Finding ◽  
Points Of View ◽  
Iteration Processes

A survey of some modifications based on the classic Newton’s and the higher order Newton-like root finding methods for complex polynomials is presented. Instead of the standard Picard’s iteration several different iteration processes, described in the literature, which we call nonstandard ones, are used. Kalantari’s visualizations of root finding process are interesting from at least three points of view: scientific, educational, and artistic. By combining different kinds of iterations, different convergence tests, and different colouring we obtain a great variety of polynomiographs. We also check experimentally that using complex parameters instead of real ones in multiparameter iterations do not destabilize the iteration process. Moreover, we obtain nice looking polynomiographs that are interesting from the artistic point of view. Real parts of the parameters alter symmetry, whereas imaginary ones cause asymmetric twisting of polynomiographs.

scholarly journals Computational geometry as a tool for studying root-finding methods

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1019-1027 ◽  
Author(s):  
Ivan Petkovic ◽  
Lidija Rancic
Keyword(s):  
Computational Geometry ◽  
Basins Of Attraction ◽  
Dynamic Study ◽  
Third Order ◽  
Root Finding ◽  
Mathematical Visualization ◽  
Order Root ◽  
Insight Into ◽  
Basins Of Attractions

We present an efficient method from Computational geometry, a branch of computer science devoted to the study of algorithms, for mathematical visualization of a third order root solver. For many decades the quality of iterative methods for solving nonlinear equations were analyzed only by using numerical experiments. The disadvantage of this approach is the inconvenient fact that convergence behavior strictly depends on the choice of initial approximations and the structure of functions whose zeros are sought, which often makes the convergence analysis very hard and incomplete. For this reason in this paper we apply dynamic study of iterative processes relied on basins of attraction, a new and powerful methodology developed at the beginning of the 21th century. This approach provides graphic visualization of the behavior of convergent sequences and, consequently, offers considerably better insight into the quality of applied root solvers, especially into the domain of convergence. For demonstration, we present dynamic study of one parameter family of Halley?s type introduced in the first part of the paper. Characteristics of this family are discussed by basins of attractions for various values of the involved parameter. Special attention is devoted to clusters of polynomial roots, one of the most difficult problems in the topic. The analysis of the methods and presentation of basins of attractions are performed by the computer algebra system Mathematica.

scholarly journals Abnormal higher order aberrations in anisometropic amblyopia

2020 ◽  
Author(s):  
Lingyi Zhang ◽  
Meng Liao ◽  
Wenqiu Zhang ◽  
Longqian Liu
Keyword(s):  
Visual Acuity ◽  
Pearson Correlation ◽  
Higher Order ◽  
Control Group ◽  
Third Order ◽  
Higher Order Aberrations ◽  
Regular Treatment ◽  
Anisometropic Amblyopia ◽  
Pearson Correlation Test ◽  
Order Root

Abstract Background The study compared ocular higher-order aberrations (HOAs) between monocular anisometropic amblyopia children and similar anisometropia children with normal best-corrected visual acuity (BCVA). The amblyopia eyes could not reach the standard BCVA level of their age even after several months' regular treatment of amblyopia. We tried to find if these intractable amblyopia eyes have abnormal HOAs. Methods Fifty school-aged children (5–9 years) with hyperopic anisometropia were recruited at West China hospital clinic. Each subject shall be reexamined once every three months, four consecutive reexaminations for 12 months, of which only 25 subjects with normal BCVA after wearing glasses shall be included in the control group. The rest 25 subjects were treated by glasses and six months’ patching. Their interocular difference of visual acuity was still more than or equal to LogMAR 0.2, and the eyes with poor visual acuity did not reach the normal level in the twelfth month. These subjects were included in the amblyopia group. The BCVA, HOAs (5 mm pupil diameter), and axial length were recorded for all subjects. Results There were significant differences of higher order aberrations in C (3, -3) between the amblyopia eyes and the other three groups of eyes with normal BCVA (all p < 0.05). There were significant differences of higher-order aberrations in third-order root-mean-square aberrations(RMS) between the amblyopia eyes and the low-diopter eyes from two subject groups(both p < 0.05). Compared to high-diopter eyes in the control group, there were a significantly higher C(3,-3) in all amblyopia eyes and a significantly higher third-order RMS (p = 0.040) in the moderate-to-severe amblyopia group. In the Pearson correlation test, the C(3,-3) and third order RMS demonstrated statistically significant correlations with BCVA ( (r=-0.19, p = 0.04; r = 0.37, p < 0.01). Conclusions The anisometropic amblyopia eyes had different HOAs from the eyes with normal visual acuity. The third-order aberrations were the primary abnormal higher-order aberration in amblyopia eyes.

scholarly journals A New Nonlinear Ninth-Order Root-Finding Method with Error Analysis and Basins of Attraction

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1996
Author(s):  
Sania Qureshi ◽  
Higinio Ramos ◽  
Abdul Karim Soomro
Keyword(s):  
Nonlinear Models ◽  
Efficiency Index ◽  
Basins Of Attraction ◽  
Nonlinear Phenomena ◽  
Numerical Schemes ◽  
Engineering Research ◽  
Root Finding ◽  
Order Root ◽  
The Lorenz System ◽  
Root Finding Method

Nonlinear phenomena occur in various fields of science, business, and engineering. Research in the area of computational science is constantly growing, with the development of new numerical schemes or with the modification of existing ones. However, such numerical schemes, objectively need to be computationally inexpensive with a higher order of convergence. Taking into account these demanding features, this article attempted to develop a new three-step numerical scheme to solve nonlinear scalar and vector equations. The scheme was shown to have ninth order convergence and requires six function evaluations per iteration. The efficiency index is approximately 1.4422, which is higher than the Newton’s scheme and several other known optimal schemes. Its dependence on the initial estimates was studied by using real multidimensional dynamical schemes, showing its stable behavior when tested upon some nonlinear models. Based on absolute errors, the number of iterations, the number of function evaluations, preassigned tolerance, convergence speed, and CPU time (sec), comparisons with well-known optimal schemes available in the literature showed a better performance of the proposed scheme. Practical models under consideration include open-channel flow in civil engineering, Planck’s radiation law in physics, the van der Waals equation in chemistry, and the steady-state of the Lorenz system in meteorology.

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