Pure 2 Coursework

Decimal Search

This method works by drawing a graph, seeing where it crosses the axis, then re-plotting the graph, with a smaller scale.  This is repeated until required accuracy is achieved.  I require an accuracy of four decimal places.  

The equation I shall investigate will be:

y = x4+2x3−5x2+1

I shall find the root by working out values of x and looking for a change of sign.  I can see by looking at the graph that the change of sign occurs between 3 and 4.  I shall plot a graph, to find a more accurate estimate.  

I can now see that the change of sign occurs between 3.4 and 3.5.  Again, I shall draw another graph to find a more accurate answer.  

Solution = -3.4325 ± 0.0005

f (−3.432) = -0.00561

f (−3.433) =  0.051138

Change of sign indicates a root somewhere between the two.  

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This method can fail if:

                The graph has two roots that are close together,

                The graph has a discontinuity,

                The graph touches the x axis, but does not cross it.  

E.g. A graph that has two roots close together:

y = 90x4 – 60x + 24.7

This would indicate that the graph does not cross the axis and therefore has no roots, but if the graph is sketched:

This method would therefore appear ineffective.  

Newton-Raphson Method

The basic principle of this method involves making an estimate (x0) ...

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