Using the Decimal Search/Change of Sign method, it took 32 calculations to reach a conclusive answer to 5 decimal places of 0.337735 ±0.000005.
Method: Newton Raphson
Using the Newton Raphson method, it took 4 iterations to reach a conclusive answer to 5 decimal places of 0.337735 ±0.000005.
Method: Rearrangement
The original equation has to be rearranged for this method. The rearranged equation is:
x = 3x^5 + 1
3
Using the Rearrangement method, it took 5 iterations to reach a conclusive answer to 5 decimal places of 0.337735 ±0.000005.
Conclusion
My results show that the most effective method is the Newton Raphson method as it took the least iterations/calculations. However, all three methods do have their strengths and weaknesses.
The Decimal Search/Change of Sign can not calculate the value of the function if the curve is a tangent to the x-axis. It is, arguably, the easiest to calculate even though there are more calculations to do.
The Newton Raphson method can not calculate the value of the function if the part of the curve at the integer, nearest to the root, is a maximum or minimum. It usually takes the least amount of iterations/calculations to calculate the value of the function.
The Rearrangement method can not calculate the value of the function if the gradient of the curve, at the point where the root lies, is greater than 1. More work has to be done as it took 1 more iteration/calculation than the Newton Raphson method, also have to rearrange the equation.
All these method calculations were done using a computer. If there was no computer to aid me, these methods would be harder.
Decimal Search/Change of sign would take a long time as there are always a lot of calculations (32).
The Newton Raphson method would also take a long time even though it normally has only a few calculations (4). This is because the equation has to be differentiated and then put into a formula.
The Rearrangement method would, in my opinion, be the quickest. This is because the equation just has to be rearranged, and then the numbers substituted into the equation. It also has only a few calculations (5), normally slightly more than the Newton Raphson method.
Overall, I think, without the aid of a computer, the Rearrangement method would be the quickest and most efficient way of calculating the value of the function. The Newton Raphson method would be second as there are less calculations to do, even though the method is harder. Decimal Search/Change of Sign would come last as there are too many calculations to do.
However, if a computer was available, any method would be suitable, if the correct software was available. This is because they all take roughly the same amount of time.
