newton.method {animation} | R Documentation |
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). This function provides an illustration of the iterations in Newton's method.
newton.method(FUN = function(x) x^2 - 4, init = 10, rg = c(-1, 10), tol = 0.001, interact = FALSE, control = ani.control(interval = 2), ...)
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? |
control |
control parameters for the animation; see ani.control |
... |
other arguments passed to ani.control |
The iteration goes on in this way:
x[k + 1] = x[k] - FUN(x[k]) / FUN'(x[k])
From the starting value x0, blue vertical lines and red points are plotted to show the location of the sequence of iteration values x1, x2, ...; red tangent lines are drawn to illustrate the relationship between successive iterations; the iteration values are in the right margin of the plot.
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$nmax , it's quite likely that the root is not reliable because the maximum number of iterations has been reached |
The algorithm might not converge – it depends on the starting value. See the examples below.
Yihui Xie
http://en.wikipedia.org/wiki/Newton's_method
op = par(pch = 19) # default example xx = newton.method() xx$root # solution ## Not run: # another function xx = newton.method(function(x) exp(-x) * x, rg = c(0, 10), init = 2) # 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 xx = newton.method(function(x) atan(x), rg = c(-2, 2), interact = TRUE) # HTML animation pages ani.start() newton.method(function(x) exp(-x) * x, rg = c(0, 10), init = 2, saveANI = TRUE, width = 600, height = 500) ani.stop() ## End(Not run) par(op)