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Iteration - Higher

Iteration - Higher

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Tutor: Bilal

Summary

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 xx the subject of the equation. Rename the subject of the equation xx to xn+1x_{n+1} and rename the other xx's on the other side of the equation to xnx_n. 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+3x2=0x^3+3x-2=0​​


There are different ways to rearrange the formula. This is one way.

x3+3x2=0x3+3x=2x(x2+3)=2x=2x2+3\begin {aligned}x^3+3x-2&=0 \\x^3 + 3x &= 2 \\x(x^2+3) &=2 \\x &= \frac {2} {x^2+3}\end {aligned}​​


Rename the xx's.

xn+1=2xn2+3\underline{x_{n+1} = \frac {2} {x_n^2+3}}​​



Use an iteration formula

The iteration formula can now be used to find an approximate solution to the equation. Given a starting value x0x_0​, substitute this value into the formula to get an answer. The answer is now called x1x_1. Take x1x_1 and substitute this value into the formula to obtain x2x_2. Then substitute in x2x_2 to obtain x3x_3, 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+3x2=0x^3+3x-2=0​​


using the iteration formula 

xn+1=2xn2+3x_{n+1} = \frac {2} {x_n^2+3}​​


Use the iteration formula 33 times, starting with x0=0x_0=0


Start by substituting in 00 to find x1x_1.

x1=202+3=23x_1=\frac {2}{0^2+3} = \frac 2 3 ​​


Now substitute x1=23x_1=\frac 2 3 in.

x2=2232+3=1831x_2=\frac {2}{\frac2 3 ^2+3} = \frac {18}{31} ​​


Substitute x2=1831x_2= \frac{18} {31} to find x3x_3.

x3=218312+3=19223207=0.59931...x_3= \frac {2} {\frac {18}{31}^2 + 3} = \frac {1922}{3207} = 0.59931...​​


Now that the iteration formula has been used 33 times, the answer for x3x_3 is taken to be an approximate solution to the equation.

x=0.599 (3.s.f)\underline{x= 0.599}\space (3.s.f)

​​

You can use the ANS button on the calculator to make it easy to perform the calculation. As the first xx value to substitute in is 00, type in 00 on the calculator and press ==. This will now show 00​ as the answer.

Now, enter the formula using the ANS button. Type 2ANS2+3\frac {2} {ANS^2+3} and then press ==. The answer should now show as 23\frac 2 3 which is the answer for x1x_1. Pressing == again should now bring up the answer for x2x_2, which is 1831\frac {18}{31}. 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=19223207x_3=\frac{1922}{3207}.​


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FAQs - Frequently Asked Questions

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