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.