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