Iteration
In a nutshell
Iteration means the repetition of a process. It involves rearranging an equation to obtain an iteration formula. The iteration formula is then used to find the roots or solutions to the equation.
Create an iteration formula
Take the equation to be solved, and rearrange it to make one of the values of x the subject of the equation. Rename the subject of the equation x to xn+1 and rename the other x's on the other side of the equation to xn. This is now an iteration formula which can be used to find the solutions to the original equation.
Example 1
Find an iteration formula for the equation
x3+3x−2=0
There are different ways to rearrange the formula. This is one way.
x3+3x−2x3+3xx(x2+3)x=0=2=2=x2+32
Rename the x's.
xn+1=xn2+32
Use an iteration formula
The iteration formula can now be used to find an approximate solution to the equation. Given a starting value x0, substitute this value into the formula to get an answer. The answer is now called x1. Take x1 and substitute this value into the formula to obtain x2. Then substitute in x2 to obtain x3, and so on. More iterations will bring the answer closer to the value of the actual root to the equation.
Example 2
Find the solution to the equation
x3+3x−2=0
using the iteration formula
xn+1=xn2+32
Use the iteration formula 3 times, starting with x0=0
Start by substituting in 0 to find x1.
x1=02+32=32
Now substitute x1=32 in.
x2=322+32=3118
Substitute x2=3118 to find x3.
x3=31182+32=32071922=0.59931...
Now that the iteration formula has been used 3 times, the answer for x3 is taken to be an approximate solution to the equation.
x=0.599 (3.s.f)
You can use the ANS button on the calculator to make it easy to perform the calculation. As the first x value to substitute in is 0, type in 0 on the calculator and press =. This will now show 0 as the answer.
Now, enter the formula using the ANS button. Type ANS2+32 and then press =. The answer should now show as 32 which is the answer for x1. Pressing = again should now bring up the answer for x2, which is 3118. In this way, every time you press =, the answer will be substituted into the formula, so pressing = again should give the next answer, which is x3=32071922.