Newton Method

Solution of Equations in One Variable
Newton s Method
Newton’s (or the Newton-Raphson) method is one of the most powerful and well-known numerical methods for solving a root-finding problem. It is used to obtain faster convergence than offered by other types of functional iteration. The derivation of Newton s method is based on Taylor polynomial.
Suppose that f?C^2 [a,b]. Let p_0?[a,b] be an approximation to p such that f (p_0 )?0 and |p-p_0 | is “small.” Consider the first Taylor polynomial for f(x) expanded about p and evaluated at x=p.
f(p)=f(p_0 )+(p-p_0 ) f^ (p_0 )+(p-p_0 )^2/2! f (?(p))
f(p)=0

0=f(p_0 )+(p-p_0 ) f^ (p_0 )+(p-p_0 )^2/2! f (?(p))
Newton’s method is derived by assuming that since |p-p_0 | is small, the term involving (p-p_0 )^2 is much smaller, so

0?f(p_0 )+(p-p_0 ) f^ (p_0 )
p?p_0-(f(p_0))/(f (p_0))
This sets the stage for Newton’s method, which starts with an initial approximation p_0 and generates the sequence {p_n }_(n=0)^? by

p_n=p_(n-1)-f(p_(n-1) )/(f^ (p_(n-1) ) ) for n?1 ?