Newton's method is a technique for solving equations of the form
f(x)=0 by successive approximation. The idea is to pick an
initial guess x0 such that f(x0) is reasonably close to 0.
We then find the equation of the line tangent to y=f(x) at
x=x0 and follow it back to the x axis at a new (and
improved!) guess x1. The formula for this is x1=x0−f(x0)f′(x0).
We then find the equation of the line
tangent to y=f(x) at x=x1 and follow it back to the x axis
to get a new (and improved!) guess x2 from the formula x2=x1−f(x1)f′(x1).
We keep on refining our guesses until we are close enough for
whatever application we have in mind. In general, we have the
recursive formula
xn+1=xn−f(xn)f′(xn)
In typical situations, Newton's method homes in on the answer extremely quickly, roughly
doubling the number of decimal points in each round. So if your
original guess is good to one decimal place, 5 rounds later you
will have an answer good to 30+ digits.