Iteration - Mathematics A-Level Revision

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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:

xn+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 Xn+1 = 1 + 11 xn - 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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