The Newton-Raphson Method or simply Newton's Method is a numerical method for estimating the roots of a polynomial f(x). This method is derived from the Taylor's Series of f(x). Given a continuous function f(x), recall that the Taylor Series of f(x) is as follows:
f(x) = f(x0) ( x – x0 )0 / 0! + f '(x0) ( x – x0 )1 / 1! + f ''(x0) ( x – x0 )2 / 2! + . . . + f n(x0) ( x – x0 )n / n!
where
n! = n ( n – 1 ) ( n – 2 ) . . . 3 . 2 . 1
0! = 1
Observe that
1 / n! ® 0 as n
® 
In addition,
if | x – x0 | < 1, then ( x – x0 )n
® 0 as n ®
Therefore,
f ''(x0) ( x – x0 )2 / 2! + . . . +
f n(x0) ( x – x0 )n / n!
®
Hence, we can drop those terms leaving us with
f(x) = f(x0) ( x – x0 )0 / 0! + f '(x0) ( x – x0 )1 / 1! = 0
Simplifying and solving for x, we get
f(x0) + f '(x0) (x – x0) = 0
x = x0 – f(x0) / f '(x0)
Recall that f '(x0) gives the slope of the line tangent to f(x) at x0. Let x0 the initial estimate of a root. Graphically, the value of x at which the line tangent to f(x) at x0 crosses the x–axis becomes the next approximation of that root. Newton's Method converges faster than either the Bisection Method or Regula–Falsi Method if it converges at all.
Difficulties with Newton's Method:
EXAMPLE: Consider f(x) = x3 + 3x – 5, where [ a = 1, b = 2 ] and DOA = 0.001.
| i | x | f(x) | f '(x) |
|---|---|---|---|
| 1 | 1 | – 1 | 6 |
| 2 | 1.16666666666667 | 0.0879629629629637 | 7.08333333333333 |
| 3 | 1.15424836601307 | 0.000537834590739195 | 6.99686787133154 |
PROBLEM: Develop an algorithm, expressed as a NSD, that will find an estimate of the first positive root of a given polynomial f(x) within a certain degree of accuracy DOA using the Newton's Method. Determine an initial estimate of the first positive root within one unit interval. Test for the possible encounter of a minimum or maximum point (or a point of inflection in the case of non–polynomials). What do you have to do to get out of either predicament? Use Horner's Method for evaluating f(x).