Demonstration of the Newton-Raphson method for root-finding
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, col.lp = c("blue", "red", "red"), main, xlab, ylab, ...)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 |
... |
other arguments passed to |
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.
A list containing
root |
the root found by the algorithm |
value |
the value of |
iter |
number of
iterations; if it is equal to |
The algorithm might not converge – it depends on the starting value. See the examples below.
Yihui Xie
For more information about Newton's method, please see http://en.wikipedia.org/wiki/Newton's_method
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