newton.method()

Demonstration of the Newton-Raphson method for root-finding

Yihui Xie & Lijia Yu 2017-04-04

This function provides an illustration of the iterations in Newton’s method.

Newton’s method (also known as the Newton-Raphson method or the Newton-Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function f(x).

The iteration goes on in this way:

$$x_{k + 1} = x_{k} - \frac{FUN(x_{k})}{FUN'(x_{k})}$$

From the starting value \(x_0\), vertical lines and points are plotted to show the location of the sequence of iteration values \(x_1, x_2, \ldots\); tangent lines are drawn to illustrate the relationship between successive iterations; the iteration values are in the right margin of the plot.

library(animation)
ani.options(interval = 1, nmax = 50)
par(pch = 20)

## default example
xx = newton.method()

plot of chunk demo-a

xx$root  # solution

## [1] 2

## take a long long journey
newton.method(function(x) 5 * x^3 - 7 * x^2 - 40 * x + 100, 7.15, c(-6.2, 7.1))

plot of chunk demo-b

## another function
ani.options(interval = 0.5)
xx = newton.method(function(x) exp(-x) * x, rg = c(0, 10), init = 2)

plot of chunk demo-c

## does not converge!
xx = newton.method(function(x) atan(x), rg = c(-5, 5), init = 1.5)

plot of chunk demo-d

xx$root  # Inf

## [1] Inf

## interaction: use your mouse to select the starting point
if (interactive()) {
  ani.options(interval = 0.5, nmax = 50)
  xx = newton.method(function(x) atan(x), rg = c(-2, 2), interact = TRUE)
}