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Mathematics
23 June, 08:45
In laymans terms, what are iterations in maths
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Zoie Gonzales
23 June, 09:12
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Iteration
Iteration is a way of solving equations. You would usually use iteration when you cannot solve the equation any other way.
An iteration formula might look like the following:
x n+1 = 2 + 1
xn.
You are usually given a starting value, which is called x0. If x0 = 3, for example, you would substitute 3 into the original equation where it says xn. This will give you x1. (This is because if n = 0, x1 = 2 + 1/x0 and x0 = 3).
x1 = 2 + 1/3 = 2.333 333 (by substituting in 3).
To find x2, substitute the value you found for x1.
x2 = 2 + 1 / (2.333 333) = 2.428 571
Repeat this until you get an answer to a suitable degree of accuracy. This may be about the 5th value for an answer correct to 3s. f. In this example, x5 = 2.414 ...
Example
a) Show that x = 1 + 11 x - 3
is a rearrangement of the equation x² - 4x - 8 = 0.
b) Use the iterative formula X n+1 = 1 + 11
x n - 3
together with a starting value of x1 = - 2 to obtain a root of the equation x² - 4x - 8 = 0 accurate to one decimal place.
a) multiply everything by (x - 3):
x (x - 3) = 1 (x - 3) + 11
so x² - 3x = x + 8
so x² - 4x - 8 = 0
b) x1 = - 2
x2 = 1 + 11 (substitute - 2 into the iteration formula)
-2 - 3
= - 1.2
x3 = 1 + 11 (substitute - 1.2 into the above formula)
-1.2 - 3
= - 1.619
x4 = - 1.381
x5 = - 1.511
x6 = - 1.439
x7 = - 1.478
therefore, to one decimal place, x = 1.5.
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