In the derivation of Newton’s method, to determine the formula for i+1 , the function ( ) is approximated using a first-order Taylor approximation centered at i . This problem investigates what...

In the derivation of Newton’s method, to determine the formula for


_i+1
, the function
() is approximated using a first-order Taylor approximation centered at


_i
. This problem investigates what happens when you try to use a second-order Taylor approximation.

(a) Approximating
() using a second-order Taylor approximation centered at


_i
, what is the resulting formula for


_i+1
? Note your formula will have a ± in it.

(b) In theory, a second-order Taylor approximation should be more accurate than a first-order Taylor approximation (at least when you are close to the solution). However, the formula in part (a) has several unpleasant complications that Newton’s method doesn’t have. Identify two of them.

(c) Given that


_i+1

is close to


_i
, what choice should be made for the ± in part (a)?

(d) One way to avoid the complications considered in part (b) is to note that the Taylor approximation used in part (a) contains a term of the form (

_i+1

−


_i
)². Explain why this can be approximated with −(

_i+1

−


_i
)(

_i
)/ (

_i
). If this is done, what is the resulting formula for


_i+1
? Note that the formula you are deriving is known as Halley’s method, and it is an example of a third-order method.