This coursework is about finding the roots of equations by numerical methods.

C3 Coursework - Numerical Solution to Equations

This coursework is about finding the roots of equations by numerical methods.

I am going to use three different methods to solve different equations. First of all, one root should be found successfully by using three different methods. Then, error bounds should be given and shown by graphic. A failure example is given and explained.

Change of Sign Method

I need to use an equation and use Autograph to get a rough interval and do a search to find intervals that show a change of sign.

Here is an example x³–2x²–3x+4=0

Solve x³–2x²–3x+4=0

Here is the table of values

This shows that there are three intervals containing roots:

(-2, -1), [1,2] and (2, 3)

Then carry out a decimal search in one of the identified intervals to find that root to the desired level of accuracy.

I use the interval (2, 3)

The change in sign tells us there is a root in the range (2.5, 2.6)

Now use decimal search and Excel within the interval [2.5, 2.6]

There is a root in the range (2.56, 2.57)

Now use decimal search and Excel within the interval [2.560, 2.570]

There is a root in the range (2.561, 2.562)

Now use decimal search and Excel within the interval [2.5610, 2.5620]

There is a root in the range (2.5615, 2.5616)

Now use decimal search and Excel within the interval [2.5615, 2.5616]

There is a root in the range (2.56155, 2.56156)

Error bounds

I saw from my last table in Excel that the root lies in range (2.56155, 2.56156)

So the solution bounds are 2.56155 and 2.56156

I can also write the solution bounds as 2.561555 ± 0.000005. These are the error bounds.

Approximate value of root

The root is 2.5615 to 4 decimal places.

Illustrate the method graphically

The fact that f(2) is negative and f(3) ...