Solving equations by iterative methods (3) – the Newton-Raphson method

2019 ◽  
pp. 92-93
Author(s):  
John Bird
Keyword(s):  
Iterative Methods ◽  
Solving Equations ◽  
Newton Raphson ◽  
Raphson Method

scholarly journals Fractional Newton–Raphson Method Accelerated with Aitken’s Method

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 47
Author(s):  
A. Torres-Hernandez ◽  
F. Brambila-Paz ◽  
U. Iturrarán-Viveros ◽  
R. Caballero-Cruz
Keyword(s):  
Fractional Derivative ◽  
Iterative Methods ◽  
Fractional Integral ◽  
Order Of Convergence ◽  
Fractional Operators ◽  
Speed Of Convergence ◽  
Newton Raphson ◽  
Raphson Method

In the following paper, we present a way to accelerate the speed of convergence of the fractional Newton–Raphson (F N–R) method, which seems to have an order of convergence at least linearly for the case in which the order α of the derivative is different from one. A simplified way of constructing the Riemann–Liouville (R–L) fractional operators, fractional integral and fractional derivative is presented along with examples of its application on different functions. Furthermore, an introduction to Aitken’s method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, in order to finally present the results that were obtained when implementing Aitken’s method in the F N–R method, where it is shown that F N–R with Aitken’s method converges faster than the simple F N–R.

Comparative Study of Iterative Methods for Solving Non-Linear Equations

2021 ◽  
Vol 23 (07) ◽  
pp. 858-866
Author(s):  
Gauri Thakur ◽  
◽  
J.K. Saini ◽  
Keyword(s):  
Numerical Analysis ◽  
Iterative Methods ◽  
Linear Equations ◽  
Bisection Method ◽  
Practical Applications ◽  
Non Linear ◽  
False Position ◽  
Newton Raphson ◽  
Raphson Method ◽  
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.

scholarly journals Comparative Study of Various Iterative Numerical Methods for Computation of Approximate Root of the Polynomials

2021 ◽  
Vol 11 (2) ◽  
pp. 6-11
Author(s):  
Dr. Roopa K M ◽  
◽  
Venkatesha P ◽  
Keyword(s):  
Numerical Methods ◽  
Comparative Study ◽  
Iterative Methods ◽  
Numerical Comparison ◽  
Polynomial Equations ◽  
Chebyshev Method ◽  
Newton Raphson ◽  
Raphson Method ◽  
Iterative Numerical Methods

The aim of this article is to present a brief review and a numerical comparison of iterative methods applied to solve the polynomial equations with real coefficients. In this paper, four numerical methods are compared, namely: Horner’s method, Synthetic division with Chebyshev method (Proposed Method), Synthetic division with Modified Newton Raphson method and Birge-Vieta method which will helpful to the readers to understand the importance and usefulness of these methods.

The solution of algebraic equations on the EDSAC

1952 ◽  
Vol 48 (2) ◽  
pp. 255-270 ◽  
Author(s):  
R. A. Brooker
Keyword(s):  
Iterative Methods ◽  
Algebraic Equations ◽  
Newton Raphson ◽  
Raphson Method

AbstractThis paper is an account of the methods that have been used with the EDSAC for the solution of algebraic equations. Three repetitive or iterative methods are examined: Bernoulli's method, the root-squaring method, and the Newton-Raphson method. Experience with the EDSAC has shown that, as in hand computing, quadratically convergent methods are to be preferred to those less rapidly convergent. In particular, the Newton-Raphson method has proved the most useful. Several examples are given in the appendix.

Axisymmetric flow near the critical point on the moving boundary

2010 ◽  
Vol 7 ◽  
pp. 182-190
Author(s):  
I.Sh. Nasibullayev ◽  
E.Sh. Nasibullaeva
Keyword(s):  
Fluid Flow ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Axial Velocity ◽  
Moving Boundary ◽  
Axisymmetric Flow ◽  
Boundary Motion ◽  
Newton Raphson ◽  
Raphson Method

In this paper the investigation of the axisymmetric flow of a liquid with a boundary perpendicular to the flow is considered. Analytical equations are derived for the radial and axial velocity and pressure components of fluid flow in a pipe of finite length with a movable right boundary, and boundary conditions on the moving boundary are also defined. A numerical solution of the problem on a finite-difference grid by the iterative Newton-Raphson method for various velocities of the boundary motion is obtained.

A Generalized Compositional Approach for Reservoir Simulation

1983 ◽  
Vol 23 (05) ◽  
pp. 727-742 ◽  
Author(s):  
Larry C. Young ◽  
Robert E. Stephenson
Keyword(s):  
Enhanced Recovery ◽  
Gas Condensate ◽  
Reservoir Modeling ◽  
Pressure Equation ◽  
Compositional Modeling ◽  
Fluid Property ◽  
Black Oil ◽  
Fluid Properties ◽  
Newton Raphson ◽  
Raphson Method

A procedure for solving compositional model equations is described. The procedure is based on the Newton Raphson iteration method. The equations and unknowns in the algorithm are ordered in such a way that different fluid property correlations can be accommodated leadily. Three different correlations have been implemented with the method. These include simplified correlations as well as a Redlich-Kwong equation of state (EOS). The example problems considered area conventional waterflood problem,displacement of oil by CO, andthe displacement of a gas condensate by nitrogen. These examples illustrate the utility of the different fluid-property correlations. The computing times reported are at least as low as for other methods that are specialized for a narrower class of problems. Introduction Black-oil models are used to study conventional recovery techniques in reservoirs for which fluid properties can be expressed as a function of pressure and bubble-point pressure. Compositional models are used when either the pressure. Compositional models are used when either the in-place or injected fluid causes fluid properties to be dependent on composition also. Examples of problems generally requiring compositional models are primary production or injection processes (such as primary production or injection processes (such as nitrogen injection) into gas condensate and volatile oil reservoirs and (2) enhanced recovery from oil reservoirs by CO or enriched gas injection. With deeper drilling, the frequency of gas condensate and volatile oil reservoir discoveries is increasing. The drive to increase domestic oil production has increased the importance of enhanced recovery by gas injection. These two factors suggest an increased need for compositional reservoir modeling. Conventional reservoir modeling is also likely to remain important for some time. In the past, two separate simulators have been developed and maintained for studying these two classes of problems. This result was dictated by the fact that compositional models have generally required substantially greater computing time than black-oil models. This paper describes a compositional modeling approach paper describes a compositional modeling approach useful for simulating both black-oil and compositional problems. The approach is based on the use of explicit problems. The approach is based on the use of explicit flow coefficients. For compositional modeling, two basic methods of solution have been proposed. We call these methods "Newton-Raphson" and "non-Newton-Raphson" methods. These methods differ in the manner in which a pressure equation is formed. In the Newton-Raphson method the iterative technique specifies how the pressure equation is formed. In the non-Newton-Raphson method, the composition dependence of certain ten-ns is neglected to form the pressure equation. With the non-Newton-Raphson pressure equation. With the non-Newton-Raphson methods, three to eight iterations have been reported per time step. Our experience with the Newton-Raphson method indicates that one to three iterations per tune step normally is sufficient. In the present study a Newton-Raphson iteration sequence is used. The calculations are organized in a manner which is both efficient and for which different fluid property descriptions can be accommodated readily. Early compositional simulators were based on K-values that were expressed as a function of pressure and convergence pressure. A number of potential difficulties are inherent in this approach. More recently, cubic equations of state such as the Redlich-Kwong, or Peng-Robinson appear to be more popular for the correlation Peng-Robinson appear to be more popular for the correlation of fluid properties. SPEJ p. 727

scholarly journals Correction to: Efficient analysis of welding thermal conduction using the Newton–Raphson method, implicit method, and their combination

2020 ◽  
Author(s):  
Zhongyuan Feng ◽  
Ninshu Ma ◽  
Wangnan Li ◽  
Kunio Narasaki ◽  
Fenggui Lu
Keyword(s):  
Thermal Conduction ◽  
Implicit Method ◽  
Newton Raphson ◽  
Raphson Method

A Correction to this paper has been published: https://doi.org/10.1007/s00170-020-06437-w

scholarly journals Comparing Different Approaches for Solving Large Scale Power-Flow Problems With the Newton-Raphson Method

IEEE Access ◽  
2021 ◽  
Vol 9 ◽  
pp. 56604-56615
Author(s):  
Manolo D'orto ◽  
Svante Sjoblom ◽  
Lung Sheng Chien ◽  
Lilit Axner ◽  
Jing Gong
Keyword(s):  
Power Flow ◽  
Large Scale ◽  
Flow Problems ◽  
Newton Raphson ◽  
Raphson Method
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