Solution of equations by numerical methods Essay
Solution of equations by numerical methods
The root I was looking for was either 1. 8708(5sf. ) or -1. 8708(5sf. ). Using the Newton-Rapson method I expected to find one of the above-mentioned roots. Nevertheless there was no convergence revealed. Instead the numbers were going all over the place and were dramatically getting bigger all the time. At last the calculator was not able to handle the number; an error message was encountered. The existence of the discontinuity prevented the method from working and it has failed.
Comparison of the three methods
To compare the three methods I have decided to use the equation I used for the decimal search method, which was Y = 2X3 – 5X -2. I have already shown how the roots of this equation can be found by using the decimal search method. I will apply the rearrangement and the Newton-Raphson method to find the root between points [-1,0]. We know that there is the existence of a root between these two points as there is a change of sign. From all the methods I used, the rearrangement method was the easiest to use once the correct rearrangement is done. Short and easy calculations were carried out using the iterative formulas.
In some cases it is hard to find the different roots of an equation, as different rearrangements has to be done. g'(x) also has to lie between -1 and 1, or else the value of x converges to an unwanted root, or doesn’t converge to anything at all. The Newton-Rapson method was fairly easy as well. At times the Newton-Raphson method might be laborious, as long differentiations has to be carried out. While using excel I had to insert long formulas when applying this method. This problem also happens when using a calculator. On the other hand the decimal search method takes a long time.
Finding the root of an equation to three decimal points using the decimal search method considerably takes a long time and is tedious. Speed of convergence By using the Newton-Rapson method I was able to find the roots that I was looking for very rapidly. While comparing each method using the same equation it only took four steps to get the estimated root, compared to seven for that of the rearrangement method. As it is stated above the rearrangement method has a fairly fast convergence. Changing the equation f(x) into the form g (x) is a crucial point. Any error will result in failure to find the root.
Sometimes it could be very long depending on the equation that we are trying to find the root for. The decimal search method takes a long time as it is mentioned above. Having to keep on splitting the boundaries of a root in order to reach to the closest point to the precise root is a long process. If we were trying to find a root to five or more decimal places it would be laborious and cumbersome. Although all the three methods have their own disadvantages, if carefully applied to an appropriate equation they would give a good estimation of the root(s) of an equation to a limited degree of accuracy.
Subject: Alternative solutions,
University/College: University of Arkansas System
Type of paper: Thesis/Dissertation Chapter
Date: 13 July 2017
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