Comparative Study of Iterative Methods for Solving Non-Linear Equations

In numerical analysis, methods for finding roots play a pivotal role in the field of many real and practical applications. The efficiency of numerical methods depends upon the convergence rate (how fast the particular method converges). The objective of this study is to compare the Bisection method, Newton-Raphson method, and False Position Method with their limitations and also analyze them to know which of them is more preferred. Limitations of these methods have allowed presenting the latest research in the area of iterative processes for solving non-linear equations. This paper analyzes the field of iterative methods which are developed in recent years with their future scope.

A Comparative Study among New Hybrid Root Finding Algorithms and Traditional Methods

In this paper, we propose a novel blended algorithm that has the advantages of the trisection method and the false position method. Numerical results indicate that the proposed algorithm outperforms the secant, the trisection, the Newton–Raphson, the bisection and the regula falsi methods, as well as the hybrid of the last two methods proposed by Sabharwal, with regard to the number of iterations and the average running time.

An Iterative Hybrid Algorithm for Roots of Non-Linear Equations

Finding the roots of non-linear and transcendental equations is an important problem in engineering sciences. In general, such problems do not have an analytic solution; the researchers resort to numerical techniques for exploring. We design and implement a three-way hybrid algorithm that is a blend of the Newton–Raphson algorithm and a two-way blended algorithm (blend of two methods, Bisection and False Position). The hybrid algorithm is a new single pass iterative approach. The method takes advantage of the best in three algorithms in each iteration to estimate an approximate value closer to the root. We show that the new algorithm outperforms the Bisection, Regula Falsi, Newton–Raphson, quadrature based, undetermined coefficients based, and decomposition-based algorithms. The new hybrid root finding algorithm is guaranteed to converge. The experimental results and empirical evidence show that the complexity of the hybrid algorithm is far less than that of other algorithms. Several functions cited in the literature are used as benchmarks to compare and confirm the simplicity, efficiency, and performance of the proposed method.

Theoretical Study of the Reverse Roll Coating of Non-Isothermal Magnetohydrodynamics Viscoplastic Fluid

This article describes the development of a mathematical model of the reverse roll coating of a thin film for an incompressible non-isothermal magnetohydrodynamics (MHD) viscoplastic fluid as it passes through a small gap between two rolls rotating reversely. The equations of motion required for the fluid added to the web are constructed and simplified using the lubrication approximation theory (LAT). Analytical results are obtained for the velocity profile, pressure gradient, and temperature distribution. The pressure distributions and flow rate are calculated numerically using the trapezoidal rule and regular false position method, respectively. Some of these results are presented graphically, while others are shown in a tabular form. From the present analysis, it has been observed that the magnitude of pressure distributions increases by increasing the value of the involved parameters. It is worth mentioning that the velocities ratio and Brickman’s number are controlling parameters for the temperature distributions. The results indicate the strong effectiveness of the viscoplastic parameter and velocities ratio for the velocity and pressure distributions. It is also concluded that the coating of Casson material has been remarkably affected by the magnetohydrodynamics effects.

Mathematical Philology in the Treatise on Double False Position in an Arabic Manuscript at Columbia University

This article examines an Arabic mathematical manuscript at Columbia University’s Rare Book and Manuscript Library (or. 45), focusing on a previously unpublished set of texts: the treatise on the mathematical method known as Double False Position, as supplemented by Jābir ibn Ibrāhīm al-Ṣābī (tenth century?), and the commentaries by Aḥmad ibn al-Sarī (d. 548/1153–4) and Saʿd al-Dīn Asʿad ibn Saʿīd al-Hamadhānī (12th/13th century?), the latter previously unnoticed. The article sketches the contents of the manuscript, then offers an editio princeps, translation, and analysis of the treatise. It then considers how the Swiss historian of mathematics Heinrich Suter (1848–1922) read Jābir’s treatise (as contained in a different manuscript) before concluding with my own proposal for how to go about reading this mathematical text: as a witness of multiple stages of a complex textual tradition of teaching, extending, and rethinking mathematics—that is, we should read it philologically.

Penentuan Tingkat-Tingkat Energi Vibrasi Molekul Hidrogen Pada Keadaan Elektronik Dasar Menggunakan Potensial Morse

Pendekatan Born-Oppenheimer diterapkan untuk menghitung tingkat energi vibrasi keadaan dasar molekul hidrogen. Persamaan Schrodinger untuk inti atom diselesaikan dengan menggunakan metode semi-klasik, di mana inti atom diasumsikan bergerak secara klasik dalam sumur potensial dan energi vibrasi ditentukan dengan menerapkan aturan kuantisasi kuantum. Potensial yang digunakan pada penelitian adalah potensial Morse. Dalam penelitian ini, tingkat energi vibrasi dihitung dengan metode numerik, yaitu metode integrasi Simpson dan metode regula falsi. 15 Tingkat energi vibrasi dari molekul H2 diperoleh dan dibandingkan dengan data hasil eksperimen. Perbandingan ini mengindikasikan pendekatan yang digunakan pada penelitian ini memberikan hasil yang sangat akurat pada tingkat energi vibrasi yang relatif rendah (0≤n≤4), dengan kesalahan kurang dari 0,7%, dan untuk 5≤n≤8 dengan kesalahan maksimum 7,3%. Keakuratan menurun ketika tingkat energi vibrasi meningkat. Secara khusus, untuk n = 13 dan n = 14, kesalahan meningkat secara signifikan, menunjukkan gagalnya pendekatan ini untuk tingkat energi vibrasi yang relatif tinggi, khususnya untuk dua tingkat energi ini. Born-Oppenheimer approximation was applied to calculate vibrational energy levels of ground state of Hydrogen molecule. The Schrodinger equation for the nuclei was solved using a semi-classical method, in which the nuclei are assumed to move classically in a potential well and the vibrational energies are determined by applying the quantum mechanical quantization rules. Potential used in this research was the Morse potential. Here, vibrational energy levels of the molecule were calculated using numerical methods, i.e. Simpson integration method and false position method. 15 Vibrational energy levels of the H2 molecule were obtained and compared to the corresponding results from experiments. The comparison indicated that the approximation used in this research yielded very accurate results for relatively low vibrational levels (0≤n≤4), with errors being less than 0.7% and for 5≤n≤8 with maximum of 7.3% errors. The accuracy decreased as the vibrational levels increased, as expected. In particular, for n=13 and n=14, errors significantly increased, indicating the breakdown of the approximation for relatively high vibrational levels, in particular for these two energy levels. Keywords: Hydrogen Molecule; Morse Potential; Born-Oppenheimer Approximation; Simpson Method; False Position Method

Obtenção de zeros de funções utilizando Python

Numerical Methods are very important in Engineering because many real problems have complicated mathematical models that are difficult to be solved analytically. Thus, the methods of resolution for several problems that are studied in the discipline of Computational Numerical Methods, as well as in the discipline of Algorithms, are indispensable for the formation of a future Engineer. Among the several numerical methods that exist, the following are the methods for obtaining zeros of functions: Bisection, False Position and Newton-Raphson. The Bisection method consists of defining the range containing a root and, using the arithmetic mean, dividing it until the desired precision is reached. In the case of the False Position method, the weighted arithmetic mean is used to obtain the approximate root. Finally, although Newton-Raphson's method has faster convergence than the others, the drawback of this method is the need to use the derivative of the studied function. Thus, in some cases, this method may be impracticable. In this work, the methods mentioned will be implemented in the Python programming language. In this work, the mentioned methods are implemented in the Python programming language, in order to strengthen programming knowledge in the formation of Engineers, as well as to emphasize the importance of applying numerical methods in practical problems of various engineering areas.

A Strictly Defined Orthogonal Global Task Coordinate Frame and its Contouring Control Application on Biaxial Systems

In this paper, a strictly defined new orthogonal global task coordinate frame (NGTCF) based on the false position method is proposed for precision contouring control of biaxial systems. In contrast to the existed global task coordinate frame (GTCF), the value of the normal coordinate in NGTCF directly represents the contour error, rather than the first-order approximation. Moreover, different from the conventional GTCF just suitable for contours with explicit shape functions, the proposed NGTCF can be utilized in various complex contours. The false position method is adopted to calculate the curve coordinates of actual points in NGTCF. Then an adaptive robust controller (ARC) is designed to deal with the effects of strong coupling of the system dynamics in the task space and modeling uncertainties. The proposed NGTCF-based ARC contouring control strategy is tested on a linear motor driven biaxial industrial gantry. Experiments under different contouring tasks with high-speed and large-curvature are conducted to verify the effectiveness of the proposed method, and the experimental results confirm that the excellent contouring performance of the proposed approach can be achieved.