Solution of equations by numerical methods Introduction In this coursework I am going to use numerical methods to solve equations that can’t be solved algebraically, for example if the largest power of the equation was 3 or over , we would have to use the following methods: the change in sign method, Newton-Raphson method and the method of rearranging f(x) =0 in the form x = g(x). I am going to use an equation which cannot be factorised or solved using the formula x = -b . 2a Change of sign method My equation is -6×3 +9×2 +5x-6. My equation has four roots I am going to use change of sign to find one of them.
For the change in sign method I am going to do a decimal search, i. e. I have taken the negative and positive y values that are in between zero. I did my first decimal search to two decimal places and carried on the same procedure all the way up to four decimal places. This rearrangement was successful due to the gradient between [1, 2] being smaller than 1. Comparing speed of convergence After using the function x5-6x+2 for all of the three methods I can confidently say that using the decimal search method gave the faster convergence. For instance, for x3 of decimal search I got 1.4675+ 0. 0005.
For Newton Raphson I got at x3 1. 474406026 and for the rearrangement I got x3 1. 496651121. Although Newton Raphson and the rearrangement were more accurate as it is to more decimal places. Even though the decimal search method was the most efficient it took longer to do. Comparing methods in terms of ease of use with hardware and software With all three methods I used a graphical calculator as my hardware. With the decimal search method I only had to be aware of where the roots lie in the intervals to be able to use the method.
With Newton-Raphson method I just had to be able to check on the graphics calculator what the roots were to know whether it was converging towards the root or diverging from the root. This made me use the arrow buttons to move the curser to where the roots lie. On the contrary, with rearranging I had to plot the y=x and the function on the same graph and for this I had to use Microsoft excel. In conclusion in terms of ease of use with involved hardware and software; the decimal search was the most efficient method to use, then Newton- Raphson and the most involved one was the rearranging method.