The Bisection method for solving equations is accurate but quite lengthy. A quicker method, which requires differentiation, is the Newton Rhapson method.
In this method, an initial value of x, which is thought to be near the root is found. This value becomes x0.
A tangent to the curve is drawn at this point and extended to cut the x-axis. This value is x1.
This value is found using the general formula :
 |
Note that the function f(x) must be differentiable.
When the equation has more than one root, the initial value of x must be near to the root being found or the second value of x could be nearer to another root of the equation.
This process is repeated until successive differences are less than the degree of accuracy required. i.e. I x n - xn-1I < E
Example
Use the Newton-Rhapson method to solve the equation x3 = 5 to 6 decimal places.
First of all, rearrange the equation to x3 − 5 = 0
Now, find an initial value.
|
x
|
0
|
1
|
2
|
3
|
|
x3 - 5
|
-5
|
-4
|
3
|
22
|
The root is between 1 and 2 (the sign changes from negative to positive) and nearer to 2, so we will use and initial value of x0 = 1.75
Differentiate f(x) = x3 − 5
f '(x) = 3x2
Using 
x1 = 1.75 − 
= 1.7108844
This process is now repeated until the required degree of accuracy is reached.
To summarise and continue this process in a table:
|
Initial value
|
x0
|
1.75
|
x n - xn-1
|
|
1st iteration
|
x1
|
1.7108844
|
-0.0391156
|
|
2nd iteration
|
x2
|
1.7099764
|
-0.000908
|
|
3rd iteration
|
x3
|
1.7099759
|
-0.0000005
|
As I x n - xn-1I < 0.000001 the process is stopped.
The solution is x = 1.709976 (to 6 decimal places)
Non-convergence
If the first value used in the Newton-rhapson method is not chosen correctly it could mean that successive iterations will not converge on the required root. An example is shown below .
The initial value x0 is too far from the required root and the tangents can be seen to be oscillating and not converging on the root.
Other cases of non-convergence
If the initial value is a turning point, the tangent will not intersect the x-axis.
If the initial value is too far from the required root then convergence may be to another root.
Comparison of the Bisection Method and the Newton Rhapson Method
Comparing the two methods:
|
Method
|
Advantages
|
Disadvantages
|
| Bisection |
Reliable Always converges to the root |
Slow, can require many iterations |
| Newton-Rhapson |
Fast convergence Simple formula |
Requires differentiation Can converge to wrong root May not converge at all |
A third method, the secant method can also be used.
