animation (version 2.0-5)

newton.method: 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

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

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:

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

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.

References

http://en.wikipedia.org/wiki/Newton's_method

http://animation.yihui.name/compstat:newton_s_method

See Also

optim

Examples

oopt = ani.options(interval = 1, nmax = ifelse(interactive(), 
    50, 2))
par(pch = 20)

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

## 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))

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

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

## HTML animation pages
saveHTML({
    ani.options(nmax = ifelse(interactive(), 100, 2))
    par(mar = c(3, 3, 1, 1.5), mgp = c(1.5, 0.5, 0), pch = 19)
    newton.method(function(x) 5 * x^3 - 7 * x^2 - 40 * x + 100, 
        7.15, c(-6.2, 7.1), main = "")
}, img.name = "newton.method", htmlfile = "newton.method.html", 
    ani.height = 500, ani.width = 600, title = "Demonstration of the Newton-Raphson Method", 
    description = "Go along with the tangent lines and iterate.")

ani.options(oopt)