Having done the three methods on the same equation and the same root, I can now compare their merits and limitation for each method: * Change of Sign Using the change of signs method to find the root is a very slow process, you have to use the decimal search to find the intervals where the roots are. Then, using the interval bisection, it speeds up the process a little. It is easy to use, and soon converges to the root that I am looking for.

However, the root that is found using this method is just as accurate as the other two methods, but using this method, the number of steps to work the root out is 20, and is the most steps taken in all three methods. * Newton-Raphson The Newton-Raphson method gave the fastest speed of convergence, although when calculating the root using the Newton-Raphson iteration formula takes a lot of time because xn, follows one after another.

When the iterative formula is used, it can fail to find the root if the calculations show divergence (when x1 is too close to the turning point). Using the iterative formula, only 5 steps are needed to work out the root. * Rearranging method Rearranging the function of f(x) = 0 to g(x), using the iteration formula, and the staircase diagram, this converges to the root in [0, 1] but have to rearrange the formula again in order to work out the other two.

Therefore, using this method may take longer to work out all three roots in the equation. Only 6 steps are needed to work out the root. All three methods are at 6 d. p. Uses of hardware and software: In this investigation, I had used quite a range of software packaging, which enables me to do complicated equations or graph instantly. A graph-drawing package (Autograph) benefited all the methods. I was able to create the graphs for the functions and equations. This package also enables me to zoom into a root to draw tangents.

A spreadsheet software package (Excel) allowed me to customize the tables and implement formulae that helped to calculate the approximations at each value of x. On the graphics application, I could easily revise and correct the formulae to make changes where applicable. I was then able to import the formulae into the word processor and present them alongside the graphs and explanations. For hardware, I used a calculator to help me with complicated calculations to check for answers for Newton-Raphson and Rearranging formular.

However, the root that is found using this method is just as accurate as the other two methods, but using this method, the number of steps to work the root out is 20, and is the most steps taken in all three methods. * Newton-Raphson The Newton-Raphson method gave the fastest speed of convergence, although when calculating the root using the Newton-Raphson iteration formula takes a lot of time because xn, follows one after another.

When the iterative formula is used, it can fail to find the root if the calculations show divergence (when x1 is too close to the turning point). Using the iterative formula, only 5 steps are needed to work out the root. * Rearranging method Rearranging the function of f(x) = 0 to g(x), using the iteration formula, and the staircase diagram, this converges to the root in [0, 1] but have to rearrange the formula again in order to work out the other two.

Therefore, using this method may take longer to work out all three roots in the equation. Only 6 steps are needed to work out the root. All three methods are at 6 d. p. Uses of hardware and software: In this investigation, I had used quite a range of software packaging, which enables me to do complicated equations or graph instantly. A graph-drawing package (Autograph) benefited all the methods. I was able to create the graphs for the functions and equations. This package also enables me to zoom into a root to draw tangents.

A spreadsheet software package (Excel) allowed me to customize the tables and implement formulae that helped to calculate the approximations at each value of x. On the graphics application, I could easily revise and correct the formulae to make changes where applicable. I was then able to import the formulae into the word processor and present them alongside the graphs and explanations. For hardware, I used a calculator to help me with complicated calculations to check for answers for Newton-Raphson and Rearranging formular.