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Graphical Iteration

Ifis a real function then an iteration sequence of the formcan be represented graphically using the graphsandplotted on the same axes. Any intersection of the curveand the linerepresents a fixed point of

The diagram illustrates the iteration process.

Given the pointdraw a vertical line to meetat

Draw a horizontal line to meetat

Given any initial pointwe can apply the above steps repeatedly to construct the sequenceThe diagram above suggest the sequence converges to the value ofgiven by the intersection ofand

We can illustrate this for the functionThere is a fixed point at(there is also a fixed point at) .Start with

Plotand

Plotand

Plotandand so on.

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