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Summary

Iteration

​​In a nutshell

Iteration is a method of re-inputting an output into a function to get closer and closer to the root. To find the root of f(x)f(x)f(x), you let f(x)=0f(x)=0f(x)=0 and rearrange into the format x=g(x)x=g(x)x=g(x), then iterate as follows, xn=g(xn−1)x_n=g(x_{n-1})xn​=g(xn−1​). The result will either converge closer to a root or diverge further from the root.



Iterative function

To make an iterative function, the formula must be in the format x=g(x)x=g(x)x=g(x). To begin, let f(x)=0f(x)=0f(x)=0 and rearrange.


Example 1

Find three iterative functions for f(x)=x2−2x−1f(x)=x^2 -2x-1f(x)=x2−2x−1.


First, let f(x)=0f(x)=0f(x)=0


​f(x)=x2−2x−10=x2−2x−1\begin{aligned}f(x)&=x^2 -2x-1\\0&=x^2 -2x-1\end{aligned}f(x)0​=x2−2x−1=x2−2x−1​

​

Then rearrange as you wish to. There are two functions you can make by moving x2x^2x2.

​x2=2x+1\begin{aligned}x^2&=2x+1\\\end{aligned}x2​=2x+1​​​

x=2x+1‾\underline{x=\sqrt{2x+1}}x=2x+1​​​​​

​x=2+1x‾\underline{x=2+\dfrac1x}x=2+x1​​​​

Another can be made by moving −2x-2x−2x.

2x=x2−1x=x2−12‾\begin{aligned}2x&=x^2-1\\x&=\underline{\dfrac{x^2-1}{2}}\end{aligned}2xx​=x2−1=2x2−1​​​​​



Iteration: xn=g(xn−1)x_n=g(x_{n-1})xn​=g(xn−1​)

With an iterative function in the form x=g(x)x=g(x)x=g(x)​, you will find subsequent xxx values with the aim of getting closer to the root. You use an x0x_0x0​ value to find x1x_1x1​, use x1x_1x1​ to find x2x_2x2​ and so on.

xn=g(xn−1)x1=g(x0)x2=g(x1)x3=g(x2)...\begin{aligned}x_n&=g(x_{n-1})\\x_1&=g(x_0)\\x_2&=g(x_1)\\x_3&=g(x_2)\\.\\.\\.\\\end{aligned}xn​x1​x2​x3​...​=g(xn−1​)=g(x0​)=g(x1​)=g(x2​)​​​


Example 2

Use x=x2−12x=\dfrac{x^2-1}{2}x=2x2−1​ and x0=5x_0=5x0​=5 to find x1,x2,x3x_1,x_2,x_3x1​,x2​,x3​​

​​x0=5x1=52−12=12‾x2=122−12=71.5‾x3=71.52−12=2555.625‾\begin{aligned}x_0&=5\\\\x_1&=\dfrac{5^2-1}{2}=\underline{12}\\\\x_2&=\dfrac{12^2-1}{2}=\underline{71.5}\\\\x_3&=\dfrac{71.5^2-1}{2}=\underline{2555.625}\\\end{aligned}x0​x1​x2​x3​​=5=252−1​=12​=2122−1​=71.5​=271.52−1​=2555.625​​​​



Diverging and converging iteration

The previous example showed a diverging iteration: xnx_nxn​ grew increasingly larger and diverged from the root. The opposite of this is a converging iteration, where xnx_nxn​ gets closer to the root.


Example 3

Use x=2+1xx=2+\dfrac1xx=2+x1​ and x0=2x_0=2x0​=2 to find a root rounded to 3 d.p3\space d.p3 d.p.


x0=2x1=2+12=2.5x2=2+12.5=2.4x3=2+12.4=2.4167x4=2+12.4167=2.4138x5=2+12.4138=2.4143x6=2+12.4143=2.4142x7=2+12.4142=2.4142\begin{aligned}x_0&=2\\x_1&=2+\dfrac12=2.5\\\\x_2&=2+\dfrac1{2.5}=2.4\\\\x_3&=2+\dfrac1{2.4}=2.4167\\\\x_4&=2+\dfrac1{2.4167}=2.4138\\\\x_5&=2+\dfrac1{2.4138}=2.4143\\\\x_6&=2+\dfrac1{2.4143}=2.4142\\\\x_7&=2+\dfrac1{2.4142}=2.4142\\\\\end{aligned}x0​x1​x2​x3​x4​x5​x6​x7​​=2=2+21​=2.5=2+2.51​=2.4=2+2.41​=2.4167=2+2.41671​=2.4138=2+2.41381​=2.4143=2+2.41431​=2.4142=2+2.41421​=2.4142​

As 2.4142.4142.414 is repeating, you know that you have converged onto a root.

f(x)=0f(x)=0f(x)=0 when x=2.414‾\underline{x=2.414}x=2.414​​​


​​

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FAQs - Frequently Asked Questions

How do you create and iterative equation?

You let f(x) =0 and rearrange into the format x=g(x).

What are converging and diverging iterations?

Converging iterations get closer and closer to a root. Diverging iterations get further and further away.

How do you perform iteration to find a root?

You let x_n=g(x_(n-1)), and repeat until you converge onto a root.

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