This may be expressed in a graphical form (see page 8)
The process of Newton-Raphson Iteration is displayed on the graph with regards to root ‘c’. The window in the corner shows the steps taken to reach the value for the root.
The equation may be solved to find roots a, b, and c (see page 9)
For root ‘a’:
For root ‘b’:
For root ‘c’:
Error bounds for ‘c’
The value found in the spreadsheet above for root ‘c’ is 2.12842 (5 d.p.)
Because this value is to 5 decimal places, the error boundaries are as follows:
2.128415 < c < 2.128425
We can check that this is correct using the Change of Sign Method (Interval Bisection) as described above (see page 11).
The Newton-Raphson Method can, however, fail when certain equations must be solved. Below is an example:
Equation: 5 – 1/((x – 1)2 + 0.2)) = 0
Function: f(x) = 5 - 1/((x – 1)2 + 0.2))
This cannot be solved using the Newton-Raphson Method because the gradient of the tangent at the starting point is very shallow. Therefore the method makes the tangent diverge away from the root.
The function expressed in graphical form is shown on page 13.
An attempt to solve the equation using the Newton-Raphson Method is shown below:
If y = 5 – 1/((x – 1)2 + 0.2))
Then dy/dx = (50 * (x - 1)) / (5x2 – 10x + 6)2
As can be seen from this spreadsheet, it is impossible to find root ‘x’ in this case through the use of the Newton-Raphson Method for the above reasons.
The Rearrangement Method
Equation: 4 + 3x – x3 = 0
Function: f(x) = 4 + 3x – x3
Solve: 4 + 3x – x3 = 0
Draw: f(x) = 4 + 3x – x3 (see page 15)
There are two different rearrangements of 4 + 3x – x3 = 0:
Rearrangement 1
x = 4 – x3
3
and y = x
Rearrangement 2
x = 3√ 4 + 3x
and y = x
Using Rearrangement 1 draw y = x and y = g(x) (see page 16)
Using Rearrangement 2 draw y = x and y = g(x) (see page 17)
The method used to solve the two equations is shown on page 18.
The graphs showing the divergence from the root and the convergence to the root for Rearrangements 1 and 2 are shown on pages 19 and 20 respectively.
Rearrangement 1
Rearrangement 2
The equations solved:
Rearrangement 1 can clearly be seen to have failed, as no number is repeated to 5 decimal places in the ‘x’ (or ‘root’) column. The x-axis value at the intersection of the line and the curve is also far away than the root which may be seen on the original graph.
This failure when using the spreadsheet could have been predicted, as the gradient of the curve at 2 is less than –1.
The gradient of the curve at the beginning point (X1) must be between –1 and 1.
i.e. -1 < g’(x) < 1
Here it may be seen that the root numbers are the same to 5 decimal places.
Thus we know that the root is
2.19582 to 5 decimal places.
Graph showing a divergence from the root (Using Rearrangement 1), as solved in the spreadsheet at the top of page 18:
As noted briefly above, the value of g’(x) is very important in using the rearrangement method. This is less than –1 at the root ( x = 2 ) and so this method will not work.
The gradient may be found by entering the value of ‘2’ into the equation found when differentiating the equation used in Rearrangement 1:
Rearrangement 1 equation: y = 4 – x3
3
Differentiated: dy/dx = -x2
Putting x = 2 into the equation: dy/dx = -4
-4 < –1, so this is proof that Rearrangement 1 does not work.
Graph showing a convergence to the root (Using Rearrangement 2), as solved in the spreadsheet at the bottom of page 18:
The value of g’(x) at the root ( x = 2 ) is less than 1 here (as may be seen when compared to the gradient of line ‘y = x’), and so the rearrangement method works here.
The gradient may be found by entering the value of ‘2’ into the equation found when differentiating the equation used in Rearrangement 2:
Rearrangement 1 equation: y = 3√ 4 + 3x
Differentiated: dy/dx = 1 / ( 3x + 4 ) 2/3
Putting x = 2 into the equation: dy/dx = 0.215443
-1 < 0.215443 < 1, so this is proof that Rearrangement 2 does work.
Comparison of Methods
The equation used in the rearrangement method (y = 4 + 3x – x3) will be solved using the Change of Sign and Newton-Raphson methods:
Change of Sign:
Equation: 4 + 3x – x3 = 0
Function: f(x) = 4 + 3x – x3
Using Interval Bisection (see page 22)
Newton-Raphson
Equation: 4 + 3x – x3 = 0
Function: f(x) = 4 + 3x – x3
Using Newton-Raphson (see page 23)
This shows that the root is approximately 2.19582 (correct to 5 decimal places).
The three methods (Change of Sign, Newton-Raphson, and the Rearrangement Method) differ in their ease-of-use and speed.
The above examples will be used in this comparison (using the function
f(x) = 4 + 3x – x3 for the three methods).
The Change of Sign took 9 steps to find the root. This was accomplished quickly and easily through using a spreadsheet.
The Newton-Raphson method, however, took 30 steps and still was not fully finished. This is time-consuming and unnecessary.
The Rearrangement Method took 10 steps in order to prove success or express failure. It also involved the use of manual calculations, which are relatively slow when compared to the use of a spreadsheet.
Thus, the Change of Sign method was the easiest and quickest method to use.
The Hardware available plays a large part in the ease of each method. A computer helps greatly, as complex and repetitive calculations can be done easily. This is helpful in all three methods, as may be seen above.
Graphical calculators can also make the methods easier to complete. Methods that are graph-intensive, such as the Rearrangement Method, can either be completed or checked using a graphical calculator.
Other Hardware can also be used, however, these are generally large, specialised machines that are specifically designed to find the root to complex equations.
The Software available also can make different methods a lot easier. Spreadsheets (using Microsoft Excel) and Graphs (predominantly drawn using Autograph) can be incorporated into text documents (Microsoft Word) quickly and clearly, and thus reference can be made to them throughout a report. Difficult equations (such as that used in Rearrangement 2 in the Rearrangement Method) can be differentiated using Derive, which is helpful as it saves time and prevents mistakes. Therefore, the software available plays a large part in the ease of which each method can be completed.