Numerical Methods - Iteration for GCSE 9-1 and A-Level C3

In this tutorial, you will learn:

  • How to find a range of the variable, x, where a root - or a solution - of a function could exist in
  • How to turn a function into an iterative form
  • How to use iteration to find a specific root - or a solution

Finding a range for a root by change in sign method

If f(x) is a function and it changes its sign from +ve to -ve or vice versa in a certain range of x, then a root of the function exists within the same range. The following image shows it:

 

roots of a function

 

E.g.1Find a range of x in which a root for f(x) = x2 -x -3 exists.

f(x) = x2 -x -3
f(2) = 4 - 2 - 3 = -1 => -ve
f(3) = 9 - 2 - 3 = 4 => +ve
Since there is a chagne in sign in the range, 2≤ x ≤3, a root of the function exists in the same range.

Exceptions

Sometimes, there is not a change in sign, yet there will be a root in a particular region. On the other hand, a change in sign does not necessarily indicate the existance of a root within a range. The following images illustrate this:

 

exceptions for change in sign rule

 

 

Iteration

If f(x) = 0 and it can be rearranged in such a way that x = g(x), where both f(x) and g(x) are functions of x, x = g(x) is said to be in iterative form.

E.g.

x2 - x - 6 = 0

Let's make 'x' the subject of this equation; there are three possibilities:


  • x = √(6 + x)
  • x = x2- 6
  • x = 1 + 6 / x

In the above iterations, g(x) = √(6 + x) or g(x) = x2- 6 or g(x) = 1 + 6 / x.

When the above equation - x2 -x - 6 = 0 -  is written in one of the above forms with 'x' being the subject, they are said to be in an iterative form. That means, when the a value is substituted for 'x' on the right hand side the value of 'x' on the left can be obtained. Then the latter is put back in the equation to generate the other value. This goes on until, x approaches a constant value. This value is taken as a solution. Therefore, the actual iterative formula takes the following form, depending on the rearrangement.


  • xn+1 = √(6 + xn)
  • xn+1 = xn2 - 6
  • xn+1 = 1 + 6/xn

The first value of x - x0 - is taken from a range of possibilities - guesses. Now let's solve one of the above equations using iteration.


xn+1 = √(6 + xn)



  1. First, guess a range where a solution could exist. Press the button and it will provide you with one.


  2. Now fill in the value of 'x0' from one of the value you get and press the iterate button to find the solution:
    x0 =



Have you noticed the way 'x' values approach the solution?; the longer you go the better.You can now see the beauty of iteration; it helps us find the root of an a function; since we have different forms of iteration, we can use some of them to find all the roots.


Have you noticed the way 'x' values approach the solution? The longer you go the better. You can now see the beauty of iteration; it helps us to find the solution of an equation; since we have different forms of iteration, we can use each one of them to find some of the roots, if not all.

Of course, in this simple example, a quadratic function was used for illustration. Iteration is generally used for higher polynomials such as cubic functions and more complex functions.

E.g.

Find a root of the function, x3 + 4x - 3 by iteration.

f(x) = x3 + 4x - 3
f(0) = -3; f(1) = 2 => There is a change in sign between x = 0 and x = 1; therefore, a root could exist in the range.
Now, let's make the iterative formuala.
x3 + 4x - 3 = 0
x = (3 - x3)/4
xn+1 = (3 - xn3)/4
Let x0, the initial value = 0.
x1 = 0.5
x2 = 0.71875
x3 = 0.65717
x4 = 0.67904
x5 = 0.67172
x6 = 0.67422
x7 = 0.673377...

So, the root is approximately, x = 0.673 (3 d.p.).

 

 

Iteration with Microsoft Excel

The following animation shows how to use Microsoft Excel in finding a root of a function by iteration.

E.g.

Find a root of the function, f(x) = x3 + 3x - 5 by iteration.

xn+1 = (5 - xn3)/3

f(1) = -1 => -ve; f(2) = 9 => +ve;
So, there must be a root between x = 1 and x = 2. The iteration will find it.
Let x0 = 1.

 

iteration with Microsoft Excel

 

As you can see, x = 1.67 is a root to 2 d.p.

Practice is the key to mastering maths; please visit this page, for more worksheets.

Please work out the following questions to complement what you have just learnt.


  1. Rearrange x2 - 8x + 4 =0 in the form of xn+1 = 8 - 5/xn and find the solution, correct to three significant figures
  2. Rearrange x3 - 5x2 - 18 = 0 in the form of xn+1 = 18 / xn2 + 5 and find the solution, correct to three significant figures
  3. Solve x2 - 5x + 6 =0, using x = 2.5 as the initial value.


 

 

 

 

Resources at Fingertips

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level. They are organized according to these specific levels.
The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.
This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.

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