---
title: newton.method()
subtitle: Demonstration of the Newton-Raphson method for root-finding
date: '2017-04-04'
slug: newton-method
---

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.

 
```{r demo-a, cache=TRUE, interval=1}
library(animation)
ani.options(interval = 1, nmax = 50)
par(pch = 20)

## default example
xx = newton.method()
xx$root  # solution
```

```{r demo-b, cache=TRUE, interval=1}
## 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))
```

```{r demo-c, cache=TRUE, interval=0.5}
## another function
ani.options(interval = 0.5)
xx = newton.method(function(x) exp(-x) * x, rg = c(0, 10), init = 2)
```

```{r demo-d, cache=TRUE, interval=1}
## does not converge!
xx = newton.method(function(x) atan(x), rg = c(-5, 5), init = 1.5)
xx$root   # Inf
```

```{r demo-e, cache=TRUE, interval=.1, eval=FALSE}
## 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)
}
```
