Solving Equations by Iteration (Investigation)

In mathematics we learn methods to solve simple equations such as  2 x + 4 = 5 and more difficult equations such as x^{2} + 6 x + 8 = 0.  But how can we find the solution for more complex equations such as x^{4} - 2 x = 0

Guess and Check

We could guess the solution for x , substitute it into the equation then evaluate the equation to check if it is a true statement.  If it is not true we will try another value for x.

Example:

Solve x^{4} - 2 x = 0

Guess	x	=	1
Check does	1^{4} - 2	=	0
	- 1	=	0	not true

Guess	x	=	2
Check does	2^{4} - 2 \times 2	=	0
	16 - 4	=	0
	12	=	0

What is the problem with this process and what do we need to consider?  What are suitable values to try for x?

Another possible method could be to write x as a function of x  ,f x .  The equation can be rearranged as follows:  x = \frac{1}{2} x^{4}

In this case we can try a value for x, evaluate a new value forx and then check if our new value is close to the old value.

Pick an initial guess for x and call this x_0  (are there any restrictions for values of x?)

x_{1}=\frac{1}{2}x{_{0}}^{4} , evaluate for x_1

Then use x_1as the next guess for x

x_{2}=\frac{1}{2}x{_{1}}^{4} , evaluate for x_2

 

Write an algorithm for this process

Solution

• Make an initial guess for x
• Evaluate  \frac{1}{2} x_4  to calculate new value for x
• Repeat until x_{i + 1} - x_i < 0.0001
• Stop

 

Draw a flowchart for this process

Solution

 

We can use an excel spreadsheet to perform these iteration calculations

The spreadsheet table could look like this

x_i	x_i + 1
1	*
**

• What cell formula should be written in the yellow cell to calculate our new value for x?
• What cell formula should be written in the red cell to calculate our new value for x?
• Copy the formulas in the red and yellow cells to the cells below by using the drag and fill function
• What does the table look like when the first value for x tried is 1?

x_i + 1	x_1
1	0.5
0.5	0.03125
0.03125	0.0000004768371582

 

• What does the table look like when the first value for x tried is 10?

x_i + 1	x_i
10	5000
5000	312500000000000
312500000000000	4.76837E+57
4.76837E+57	2.58494E+230

Are the values of x converging (approaching a particular number) or diverging (not approaching a number - getting further from each successive value for x)?  Will the loop in our flowchart /program ever terminate?  This method of solving the equation has not been successful.

 

• Can the original equation have a different arrangement of the form x =f\left(x\right) ?   What is this?

 

The original equation can be rearranged as follows: \sqrt[4]{2x_{i}} 

Change your flowchart for this new formula

 

Set up a spreadsheet in excel using this new formula and use the process of iteration to calculate successive values of x. Fill in the cells below.

x	\sqrt[4]{2x_{i}}
1

Solution

x	\sqrt[4]{2x_{i}}
1	1.189207115
1.189207115	1.241857812
1.241857812	1.255380757
1.255380757	1.25878444
1.25878444	1.259636801
1.259636801	1.259849982
1.259849982	1.259903282
1.259903282	1.259916608
1.259916608	1.259919939

• Are the values of x converging or diverging?  This method of solving the equation has been successful. 

The solution for x correct to 4 decimal places is 1.2599

• What happens if we try a different value for the original value of x?  Lets try the value 10.

 

Solution

x	\sqrt[4]{2x_{i}}
10	2.114742527
2.114742527	1.434075029
1.434075029	1.30136903
1.30136903	1.270157613
1.270157613	1.26247243
1.26247243	1.260558411
1.260558411	1.26008036
1.26008036	1.259960876
1.259960876	1.259931006
1.259931006	1.259923539
1.259923539	1.259921672

The solution for x correct to 4 decimal places is 1.2599