Revision:Iteration

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This is a way of solving equations. An iteration formula might look like the following:

.

You are usually given a starting value, which is called x₀.

If x₀ = 3, substitute 3 into the original equation where it says x_n.

This will give you x₁.

(This is because if and ).

(by substituting in 3).

To find x₂, substitute the value you found for x₁.

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

is a rearrangement of the equation:

.

(b) - Use the iterative formula:

together with a starting value of x₁ = -2 to obtain a root of the equation:

accurate to one decimal place.

(a) multiply everything by (x - 3):

so

so

(b) x₁ = -2

(substitute -2 into the iteration formula)

(substitute -1.2 into the above formula)

therefore, to one decimal place, x = 1.5.

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