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

1992 ◽  
Vol 25 (3) ◽  
pp. 347-354 ◽  
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.

scholarly journals DEVELOPMENT OF NEW ITERATIVE SCHEME FOR SOLVING NON-LINEAR EQUATIONS

2021 ◽  
Author(s):  
Tusar singh ◽  
Dwiti Behera
Keyword(s):  
Iterative Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Iterative Scheme ◽  
Linear Equations ◽  
Quadrature Rule ◽  
Modified Newton’S Method ◽  
Newton Raphson ◽  
Raphson Method ◽  
Non Linear Equations

Within our study a special type of 〖iterative method〗^ω is developed by upgrading Newton-Raphson method. We have modified Newton’s method by using our newly developed quadrature rule which is obtained by blending Trapezoidal rule and open type Newton-cotes two point rule. Our newly developed method gives better result than the Newton’s method. Order of convergence of our newly discovered quadrature rule and iterative method is 3.

scholarly journals Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations

Algorithms ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 78
Author(s):  
Ankush Aggarwal ◽  
Sanjay Pant
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Short Note ◽  
Basin Of Attraction ◽  
Computational Cost ◽  
Initial Guess ◽  
Mathematical Functions ◽  
New Class ◽  
Root Finding ◽  
Roots Of Equations

Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is Newton’s method. However, its convergence depends heavily on the initial guess, with poor choices often leading to slow convergence or even divergence. In this short note, we seek to enlarge the basin of attraction of the classical Newton’s method. The key idea is to develop a relatively simple multiplicative transform of the original equations, which leads to a reduction in nonlinearity, thereby alleviating the limitation of Newton’s method. Based on this idea, we derive a new class of iterative methods and rediscover Halley’s method as the limit case. We present the application of these methods to several mathematical functions (real, complex, and vector equations). Across all examples, our numerical experiments suggest that the new methods converge for a significantly wider range of initial guesses. For scalar equations, the increase in computational cost per iteration is minimal. For vector functions, more extensive analysis is needed to compare the increase in cost per iteration and the improvement in convergence of specific problems.

scholarly journals On the Newton-Raphson method of Approximation

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.

scholarly journals Convergence of Newton Raphson Method and its Variants

2021 ◽  
Vol 23 (07) ◽  
pp. 1158-1164
Author(s):  
Keyword(s):  
Numerical Analysis ◽  
Convex Programming ◽  
Newton's Method ◽  
Optimization Problems ◽  
Continuous Analogue ◽  
Convergence Method ◽  
History Of ◽  
Newton Raphson ◽  
Interior Points ◽  
Raphson Method

In Numerical Analysis and various uses, including operation testing and processing, Newton’s method may be a fundamental technique. We research the history of the methodology, its core theories, the outcomes of integration, changes, they’re worldwide actions. We consider process implementations for various groups of optimization issues, like unrestrained optimization, problems limited by equality, convex programming, and methods for interior points. Some extensions are quickly addressed (non-smooth concerns, continuous analogue, Smale’s effect, etc.), whereas some others are presented in additional depth (e.g., variations of the worldwide convergence method). The numerical analysis highlights the quicker convergence of Newton’s approach obtained with this update. This updated sort of Newton-Raphson is comparatively straightforward and reliable; it’d be more probable to converge into an answer than either the upper order strategies (4th and 6th degree) or the tactic of Newton itself. Our dissertation could be about the Convergence of the Newton-Raphson Method which is a way to quickly find an honest approximation for the basis of a real-valued function g(m) = 0. The derivation of the Newton Raphson formula, examples, uses, advantages, and downwards of the Newton Raphson Method has also been discussed during this dissertation.

Optimal Power Flow Using Newton’s Method

2012 ◽  
Vol 3 (2) ◽  
pp. 167-169
Author(s):  
F.M.PATEL F.M.PATEL ◽  
◽  
N. B. PANCHAL N. B. PANCHAL
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Power Flow ◽  
Optimal Power Flow ◽  
Optimal Power
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