Method 2: The Newton-Raphson Method
The next method, which can be used to solve functions of x numerically, is the Newton-Raphson method. An example of how this method works is shown below.
The graph of f(x)=x5-6x+1 was plotted using a graph-plotting program, the result is shown below.
Using an estimate for (a.) as 0, the Newton-Raphson iterative formula was then used.
This next diagram gives some indication as to how the Newton- Raphson method actually works. What it basically does is to ‘slide’ the tangent across and is indeed sometimes known as ‘tangent sliding’.
For this method, the root can be calculated much more quickly then using the Change of Sign method. In accordance with the question the root was calculated to five significant figures, this means that the real value of x exists between 0.16655 and 0.16665, which give an error value of ± 0.00005.
This method, despite its speed in calculation a root, does not work for every function; this is illustrated below.
Method 3: Fixed Point Estimation- Rearranging the function
The third and final method for numerically solving a function of x is by rearranging an existing function of x and then using fixed-point iteration. This is illustrated for the function f(x)=x3+x-3 below.
Diagram of f(x)=x3+x-3
As is displayed graphically, the function with regards to (g) follows a ‘cobweb’ pattern. This gives rise to natural error bounds; the figure is accurate to 5 significant figures or ±0.00005.
However, this method does not always yield the required root. Since the estimation must be of a gradient less than 1, (in order for it converge to the equation of y=x), the root is not always obtained. In addition to this, as we have seen before, the method does not always function accurately when many roots are close together. Like the other methods, the plotting of a graph in advance rectifies this.
However, as stated in the question for this assessment, the aim is to find a rearrangement of the function, which does not provide the required result. The key is in the rearranging of the original function; this is displayed in the following workings.
Comparison of Methods
Using the function of f(x)=x5+3x2+x+2, which I have already used to calculate the root using the Change of Sign method, I examined calculating the same root with the other two methods.
To remind oneself, the function being solved is this:
The Newton-Raphson method
From this, it is clear that the root is correctly calculated and is clearly the same as what was previously calculated.
Fixed-Point Estimation-Rearranging the function
Therefore in conclusion, in terms of its speed, the Newton-Raphson method is highly effective in the calculation of a root. As we have seen in the previous example, both in terms of ease and rapidity, the Newton-Raphson method is highly effective. In addition, it is much harder to find a function of x for which it does not work. Unlike the rearrangement method, there is not the critical step of rearranging an equation and then the examining of it on a graph to find a suitable rearrangement. That is a rearrangement where the gradient is less than 1 or –1.
This examining on a graph is only needed once, in order to have an estimate. As well as this, the estimate need only be approximate as there is a good chance even a simple estimate will ‘home in’ on the required root. The change of sign method, although reasonably simple is not as time-effective and quick in its finding of a root.
However, like all of the methods, careful consideration must be made when there are a number of roots close together. Newton-Raphson can fail, as is shown above, so sometimes a combination of the using of a zoomed-in graph generator, the Newton-Raphson method and the careful checking with the Change of sign method can work.
Rearranging the function is another useful technique, especially when a ‘cobweb’ effect is produced, as this gives natural error bounds for the root. However, it relies on an important rearrangement of the function, in conjunction with the plotting of at least two graphs. Once this has been done, the iterative formula is used, though within my calculations did not prove as fast as the Newton-Raphson method.
As I have already mentioned, the rearrangement method, relies more heavily on suitable graphical generating software or hardware; this is therefore a clear disadvantage that it faces. However, the change of sign method requires an interval for the root and the Newton-Raphson method requires an approximation to the result. Though, as this investigation has proved clear, regardless of the method, it has been necessary to have some idea of the function before attempting calculations. This is particularly important in the case of turning points and roots close to each other.
