newton.method {animation} R Documentation

## Demonstration of the Newton-Raphson method for root-finding

### Description

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

### Usage

```newton.method(FUN = function(x) x^2 - 4, init = 10, rg = c(-1, 10), tol = 0.001,
interact = FALSE, col.lp = c("blue", "red", "red"), main, xlab, ylab, ...)
```

### Arguments

 `FUN` the function in the equation to solve (univariate), which has to be defined without braces like the default one (otherwise the derivative cannot be computed) `init` the starting point `rg` the range for plotting the curve `tol` the desired accuracy (convergence tolerance) `interact` logical; whether choose the starting point by cliking on the curve (for 1 time) directly? `col.lp` a vector of length 3 specifying the colors of: vertical lines, tangent lines and points `main, xlab, ylab` titles of the plot; there are default values for them (depending on the form of the function `FUN`) `...` other arguments passed to `curve`

### Details

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] - 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 x1, x2, …; tangent lines are drawn to illustrate the relationship between successive iterations; the iteration values are in the right margin of the plot.

### Value

A list containing

 `root ` the root found by the algorithm `value ` the value of `FUN(root)` `iter` number of iterations; if it is equal to `ani.options('nmax')`, it's quite likely that the root is not reliable because the maximum number of iterations has been reached

### Note

The algorithm might not converge – it depends on the starting value. See the examples below.

Yihui Xie

`optim`