Change of sign method - Finding a root by using change of sign method

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Change of sign method

Finding a root by using change of sign method

I will divide interval into a number to apply the change of sign method by using decimal search..

The two values are the new interval's two sides on the number axis and the roots must lie between these intervals.

i am going to use the following equation to find out a root.

it is clear that we can see the first interval lies on [1, 2]. Then i will use Excel.












In this method, I take increments in size 0.1 within the interval [1 , 2]

Work out each value of and see whether the value is positive or negative. In this case, I will use Microsoft Exel to solve it.

The diagram decimal research above shows how the sign change between [1.5,1.6]. Because of this, the interval of this equation will be [1.5,1.6].

The following graph can prove whether the interval is correct. I decide to use Autograph.

This graph has been zoom in from the first graph. It is very clear to see that the result is correct

Which is between[1.5,1.6].

I use the same method to keep doing decimal research. to work out a more accurate answer. Take increments in size 0.1 within the interval [1.5,1.6]

From the above we can see that must lie between [1.52,1.53].It can be very clear if I use graph.

I will do a better research this time by using the same method to prove my result is correct.

Now I can find a more accurate result from the research which lies between [1.521,.1522].

Here is a graph to prove the interval is right.

However, I will take 4 decimal places to improve the accuracy of the interval. Also I will use the same method again.

As we can see the root is between [1.5213,1.5214]

Same again. Autograph is used to prove my solution is right.

Error bounds

This is the process which check how the accuracy of the roots are.

From those 4 decimal search I have done so far, I can say that the answer is between 1.5213 and 1.5214. These can be improved the accuracy.
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Assume X=1.5213 f(x)=(1.5213)^3-1.5213-2=-0.00047

X=1.5214 f(x)=(1.5214)^3-1.5214-2=0.000121

Because the answer is -0.00047<0<0.000121.

So the answer must between 1.5213 and 1.5214.

However , these are not the exact answer so I have to estimate them.

In this case, X=1.5213.5, so the error bound is .

Because this is the middle point between the interval.

Fail example by using Exel

It is not guaranteed to use this method, because there still has some problems in it.

See the graph below:

As we can see the curve touches the x axis. The ...

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