History. Examples. MATLAB.
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In 1685, a root-finding algorithm called Newton's method (also known as Newton-Raphson Method) was first published as A Treatise of Algebra Both Historical and Practical, which was written by John Wallis, not Isaac Newton. Isaac Newton had described his method in 1669, but it was not published until 1711 as De analysi per aequations numero terminorum infinitas. In 1671, he also wrote De metodis fluxionum et serierum infinitarum, which was translated and published in 1736 as Method of Fluxions.
In 1960, Joseph Raphson described the Newton-Raphson method, which avoided Newton's subsitutions, in Analysis aequationum universalis. It wasn't until 1740, that Thomas Simpson published the Newton-Raphson method in current day form, which uses the derivative in the formula. The derivative part of the formula was derived from the Taylor series.
MATLAB integrates algorithm development, data
visualization, data analysis, and numeric
computation in one environment.
MATLAB is also known as The Language of
For f(x) = -x4 + 3x2 +2 convergence to the root is not always possible. When the initial guess is between x = -1 and x = 1, it will not converge to the root, because the tangent lines get trapped between x = -1 and x = 1 (fig. top left). When the initial guess is exactly 1 or -1, then the nth terms will alternate between exactly -1 and 1. An initial guess outside of those two points is guaranteed to converge (fig. top right).
Guide is part of MATLAB that is used to develop graphical user interfaces.
The function f(x) = x2 – 4 is an good example of a function that does not run into any problems with convergence. It has quadratic convergence and approximately doubles at each iterations, which makes it have few iterations and fast convergence.
Graphical User Interface
Starting with an initial point x0 or (x0,0), closer approximations to the root (r) of a function are found by finding (x0,f(x0))and using the tangent line to the function at that point. The idea of using the tangent line is that the line is a closer approximation to r of the curve. The equation for the tangent line is
y1- y0 = f’(x0) (x1- x0)
y1- f(x0) = f’(x0) (x1- x0)
Since the initial point x0 is at 0 on the y-axis, you can rewrite the equation as
0 - f(x0) = f’(x0) (x1- x0).
When that equation is solved for x1 the result is
x1 = x0 + -f(x0)
In terms of iteration number n the equation is written:
x(n+1) = xn - f(xn); n = 0, 1,2,…
The purpose of the graphical user interface for this project was to illustrate Newton's Method theoretical properties and to visually explore its key features.
The function f(x) = x2 – 2x + 1 has rather slow convergence, because its an order of convergence is one and each iteration does not double. Order one convergence is due to the double root, which is a result of f’(x)=0.
We have developed software that helps to better understand Newton’s Method and its properties. The software could be used to aid others in learning Newton’s Method.
When a function geometrically converges with an order of two, it has quadratic convergence, which means the number of accurate digits roughly doubles for each iteration. Newton’s Method is capable of this type of convergence, which can be observed through a convergence plot. A convergence plot is a plot of the error from each iteration on a log-log graph. The order of convergence is the slope of the line in the convergence plot. To read the graph from the initial guess to the root, read the graph from the right to the left.
Newton’s Method is used in a large number of applications from the production of mathematical art to weather prediction.
For f(x) = x + x4/3 the order of convergence changes depending on the root being found. When the root at x = 0 is being found, the order of convergence is one because the second derivative of the function is undefined (fig. top left). When the root at x = -1 is being found, the order of convergence is two because the second derivative of the function exists (fig. top right).
Caitlyn Davis-McDaniel and Dr. Scott Sarra Department of Mathematics, Marshall University, Huntington, WV 25755