Advanced Maths
Iteration
x2 - x - 6 = 0
Let's make 'x' the subject of this equation; there are three possibilities:
- x = √(6 + x)
- x = (x2- 6) / 5
- x = 1 + 6 / x
When the above equation - x2 + 5x - 3 = 0 - is written in one of the above form
with 'x' being the subject, they are said to be in 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)/5
- xn+1 = 1 + 6/xn
The first value of x - x0 - is taken from a range of possibilites - guesses. Now let's solve one of the
above equations using iteration.
xn+1 = √(6 + xn)
First guess a range
where a solution could exist. Press the button and it will provide you with
one.
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 to find the solution of an equation; since we
have different forms of iteration, we can use all of them to find all the
solutions.
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 all the
solutions.
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.
- Rearrange x2 - 8x + 4 =0 in the form of xn+1 = 8 - 5/xn and find the solution, correct to three significant figures
- Rearrange x3 - 5x2 - 18 = 0 in the form of xn+1 = 18 / xn2 + 5 and find the solution, correct to three significant figures
- Solve x2 - 5x + 6 =0, using x = 2.5 as the initial value.