animation (version 1.1-0)

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)

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 control$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 = 50)
op = par(pch = 20)

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

# take a long long journey
ani.options(nmax = 50)
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, nmax = 50)
xx = newton.method(function(x) exp(-x) * x, rg = c(0, 
    10), init = 2) 
# not converge!
ani.options(interval = 0.5, nmax = 50)
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
ani.options(interval = 0.5, nmax = 50)
xx = newton.method(function(x) atan(x), rg = c(-2, 
    2), interact = TRUE) 

# HTML animation pages 
ani.options(ani.height = 500, ani.width = 600, nmax = 100,
    interval = 1, title = "Demonstration of the Newton-Raphson Method",
    description = "Go along with the tangent lines and iterate.")
ani.start()
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 = "")
ani.stop()

par(op) 
ani.options(oopt)