D. Haugli, Lecturer Aer E 161 Aerospace Engineering
2/18/2005 The Newton-Raphson Method, Page 1 Iowa State University
The Newton-Raphson Method
The Newton-Raphson method is a simple, efficient method for estimating a root. The method
has the form,
()
()
n
n
nn
xf
xf
xx ′
−=
+1. (1)
Here, n
x is a guess for the value of x at iteration n, which is used to compute a new guess, 1+n
x.
The difference between the two calculated as the absolute error, a
ε
,
nn
axx −= +1
ε
. (2)
If the difference is larger than a specified convergence criterion, a
C, 1+n
x is used as the next
guess (replacing the original value of n
x), and Equation (1) is solved again. This procedure is
repeated until a
ε
becomes smaller than a
C.
Example 1. Derive an algorithm for the function,
()
0=−= qxxf p, where p is the
power and q is a constant.
Substitute
()
xf and
()
xf ′ into Equation (1), yielding:
()
()
1
1
−
+−
−= p
n
p
n
nn
xp
qx
xx . (3)
The algorithm for solving Equation (3) and finding the root is as follows:
1. Ask the user to enter the values of q and p in pqx = (how was this equation
obtained?). Ask also for a starting guess, 0=n
x, and how many digits of precision are
required, a
C.
2. Find the root
a. Calculate 1+n
x from Equation (3).
b. Calculate a
ε
from Equation (2).
c. Check for convergence (to see if the root is found).
i. If aa C<
ε
, the root is found; otherwise
ii. Set 1+
=nn xx (which makes 1=n
x the new guess) and return to Step (a).
3. Display the root and the number of times Step (2) was repeated (the number of
iterations).
