I will use the root in interval [2, 3]

Error bounds for interval [2, 3]

2.3301 is the root in this interval to 5 significant figures.

Therefore the error is 2.3301 ± 0.00005 I will now perform the change of sign test to confirm it is within these limits.

Lower limit is 2.33005 then f (2.33005) = -0.000098648

Upper limit is 2.33015 then f (2.33015) = 0.0010302

There is a change of sign which confirms root in interval is 2.3301 ± 0.00005

When does this method fail?

The Newton Raphson method does not always work, I will show this Using the equation (5x+4)1/7 =0

This function is shown below with autographs attempt to find the root using tangents to the curve starting at x= -1

The function f(x) = (5x+4)1/7

As you can see from the graph and the Values of x in the table, it was unable to find the root in the interval [-1, 0].

This is because of the gradient of the curve, where my starting value is very close to a turning point. Then rather than converging towards the root, it is diverging further and further away on the x axis, therefore overflows and is unable to find the root.

Comparison of methods

I will solve the equation which I used in the change of sign method. x5-4x+2=0

For the rearrangement method I will use the rearrangement (4x-2)1/5=0

The interval I will use for each method is [1, 2], I will use x=2 as the starting point for the rearrangement and Newton Raphson, In the change of sign method I will always go 0.1 up so the starting point is not essential, as fixed point iteration is difficult to compare with the other two methods.

The functions f(x)=x5-4x+2 , f(x)=(4x-2)1/5 , f(x)=x are shown below to allow me to do the rearrangement method on autograph.

This shows me converging onto the root in the interval [1, 2] with starting value x =2

Below is function f(x)= x5-4x+2 I will now apply the Newton Raphson method to find the same root as I found for the rearrangement method and change of sign method.

As you can see from the two previous results I got the same root as I got in the change of sign method, 1.2436 to 5 significant figures.

Comparison of the 3 methods in terms of speed of convergence

All 3 methods allowed me to obtain the same root to the same level of accuracy; this was 1.2436 to 5 significant figures. I used the same starting point for rearrangement and Newton Raphson, and used the interval [1,2] as the starting point for the change of sign.

Now that I have applied all 3 methods to one interval for one equation I can see that Newton Raphson is the fastest to converge when performing them electronically. Newton Raphson took 7 values of x until I was guaranteed 5 significant figures, where as the rearrangement method took 10 values of x. However, the change of sign method took very long compared to the other two. This was because I had to look at where the sign changed and then manually change the values of x to use myself, however it was very easy to perform as it needed no rearrangements. Newton Raphson and Rearrangement do this automatically on autograph. Overall I think Newton Raphson is the fastest method to use to solve numerical equations as it only requires one step on autograph, however rearrangement requires me to rearrange the equation first and add y=x therefore takes longer to set up.

Comparing Ease of use of Hardware and software

It’s very obvious that the hardware and software speeds up the process of working out the solutions dramatically, as manually is very time consuming and easier to make mistakes.

Autograph is very easy to use and therefore Newton Raphson and rearrangement methods are very easy to perform, as you simply have to add the equation and perform the functions required which only require very few steps and choosing the starting point. However change of sign requires more knowledge of Excel, having to input the formula yourself and choosing appropriate values for x to converge to the root. However once you know how to use excel you are able to drag down and it copies the formula for each value of x without having to retype it. For the manual calculations I was able to use a calculator, although this was time consuming, setting the starting point to “ans” allowed me to quickly get all the values of x by simply pressing the equals button. Autograph also allowed me to zoom in on roots, changing axes, etc, this helped make the process of finding roots and showing failure much easier.