On the Newton-Raphson method of Approximation

Edinburgh Mathematical Notes ◽

10.1017/s0950184300000082 ◽

1944 ◽

Vol 34 ◽

pp. 5-8

Author(s):

H. W. Richmond

Keyword(s):

Successive Approximations ◽

First Approximation ◽

Newton Raphson ◽

Image Position ◽

Raphson Method

The Method.—An equation F(x) = 0 has, a root x = r, not known exactly. From a first approximation to r, x = a, a second approximation, x = b, is obtained from the formulaFrom b a third approximation, x = c, is obtained by the same formula, and so on. The method is pointless unless the successive approximations do actually tend to r; a rule that ensures this is due to Fourier.

Download Full-text

Thomas Simpson and ‘Newton's method of approximation’: an enduring myth

The British Journal for the History of Science ◽

10.1017/s0007087400029150 ◽

1992 ◽

Vol 25 (3) ◽

pp. 347-354 ◽

Cited By ~ 14

Author(s):

Nick Kollerstrom

Keyword(s):

Newton’S Method ◽

Computer Use ◽

Newton's Method ◽

Time Warp ◽

Newton Raphson ◽

Roots Of Equations ◽

Image Position ◽

Raphson Method ◽

As If

A resurgence of interest has occurred in ‘Newton's method of approximation’ for deriving the roots of equations, as its repetitive and mechanical character permits ready computer use. If x = α is an approximate root of the equation f(x) = 0, then the method will in most cases give a better approximation aswhere f′(x) is the derivative of the function into which α has been substituted. Older books sometimes called it ‘the Newton–Raphson method’, although the method was invented essentially in the above form by Thomas Simpson, who published his account of the method in 1740. However, as if through a time-warp, this invention has migrated back in time and is now matter-of-factly placed by historians in Newton's De analysi of 1669. This paper will describe the steps of this curious historical transposition, and speculate as to its cause.

Download Full-text

Note on the Solution of Non-Linear Simultaneous Equations by Successive Approximations

Journal of the Royal Aeronautical Society ◽

10.1017/s0368393100069364 ◽

1958 ◽

Vol 62 (572) ◽

pp. 603-604 ◽

Cited By ~ 1

Author(s):

W. J. Goodey

Keyword(s):

Approximate Solution ◽

Real Root ◽

Transcendental Equation ◽

Simultaneous Equations ◽

Frequent Occurrence ◽

Successive Approximations ◽

Usual Procedure ◽

First Approximation ◽

Non Linear ◽

Newton Raphson

The Calculation of the roots of an algebraic or transcendental equation in a single unknown is a problem of frequent occurrence. For a real root the usual procedure is to obtain a first approximation to the required quantity, graphically or otherwise, and to improve this approximation by successive applications of the Newton-Raphson process. The extension of this process to the improvement of an approximate solution of a set of non-linear simultaneous equations in n unknowns is fairly obvious, but it does not seem to have received much attention in text books, although the case of two unknowns is dealt with in Ref. 2.

Download Full-text

A SECOND-ORDER NEWTON-RAPHSON METHOD FOR IMPROVED NUMERICAL STABILITY IN THE DETERMINATION OF DROPLET SIZE DISTRIBUTIONS IN SPRAYS

Atomization and Sprays ◽

10.1615/atomizspr.v16.i1.50 ◽

2006 ◽

Vol 16 (1) ◽

pp. 71-82 ◽

Cited By ~ 1

Author(s):

Meishen Li ◽

Xianguo Li

Keyword(s):

Droplet Size ◽

Numerical Stability ◽

Second Order ◽

Size Distributions ◽

Newton Raphson ◽

Droplet Size Distributions ◽

Raphson Method

Download Full-text

Axisymmetric flow near the critical point on the moving boundary

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2010.1.016 ◽

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.

Download Full-text

A Generalized Compositional Approach for Reservoir Simulation

Society of Petroleum Engineers Journal ◽

10.2118/10516-pa ◽

1983 ◽

Vol 23 (05) ◽

pp. 727-742 ◽

Cited By ~ 84

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

Download Full-text

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

The International Journal of Advanced Manufacturing Technology ◽

10.1007/s00170-020-06437-w ◽

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

Download Full-text

Fractional Newton–Raphson Method Accelerated with Aitken’s Method

Axioms ◽

10.3390/axioms10020047 ◽

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.

Download Full-text