How to prove newton's method?

As above. Thanks

1 Answer
Steve M
Jun 26, 2018

# x_(n+1) = x_n - f(x_n)/(f'(x_n)) #

Explanation:

http://www.tau.ac.il/~tsirel/dump/Static/knowino.org/wiki/Newton%27s_method.html

Suppose we seek a solution to the equation:

# f(x) = 0 #

And that we have an initial estimate #x_0# of the solution. We seek a second (hopefully more accurate) solutions, #x_1#. If #f(x)# is a smooth well behaved continuous function, then it is reasonable to assume that the point where the tangent to the curve at the point #x_0# crosses the #x#-axis is a better approximation.

The tangent to the curve at he point #(x_0,f(x_0))# has slope #f'(x_0)#, thus the equation of the tangent line is given by:

# y - f(x_0) = f'(x_0)(x-x_0) #

The point #(x_1,0)# lies on this tangent, and so it satisfies:

# 0 - f(x_0) = f'(x_0)(x_1-x_0) #

And if we rearrange for #x_1#:

# :. f'(x_0)(x_1-x_0) = -f(x_0) #

# :. x_1-x_0 = -f(x_0)/(f'(x_0)) #

# :. x_1 = x_0 - f(x_0)/(f'(x_0)) #

Leading to the general Newton's Method iterative method:

# x_(n+1) = x_n - f(x_n)/(f'(x_n)) #

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