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Firstly, the performances gained by X method and Y method will be discussed followed by the comparison between the methods developed in this research which are V, W and X.

According to the numerical results obtained in Tables 45-78, the new method which is X always need less total number of steps in order to converge to the exact solution compared to Y method. This gained is obviously seen as the tolerance getting smaller. Besides that, it can be seen the Y method achieved slightly higher accuracy than X method. Even so, the global error determined by X method still within the specified tolerance. In Problems 2.4, 2.6 and 2.8 at larger tolerance, X method have smaller global error compared to Y method whereby for Problems 2.3 and 2.5, the results demonstrate that the X method is more accurate compared to the Y methods for all tolerances.

Is there any grammatical error in these sentences? Thanks a lot..

Mister Micawber · Accepted Answer

First, the performances of the X and Y methods will be discussed, followed by a comparison of the

V, W and X methods developed in this research.

The numerical results obtained in Tables 45-78 demonstrate that the new method X always requires fewer steps in converging to the same solution as the Y method. This gain becomes more obvious as the tolerance decreases.

However, the Y method achieves slightly greater accuracy than the X method. Even so, the global error for the X method is still within the specified tolerance. At larger tolerances in Problems 2.4, 2.6 and 2.8, the X method produces a smaller global error than the Y method, while Problems 2.3 and 2.5 demonstrate that the X method is more accurate than the Y method for all tolerances.

NB: In this last paragraph, the first sentence contradicts the last sentence. Some revision is needed.

Myvi · Answer

Thanks a lot sir.. I don’t know how to repay your kindness. May *** bless you and your family..

Actually regarding to your comment, the results showed that the Y method achieved better accuracy. However in some cases, the X method outperformed Y method in a certain tolerance.

Sir if you don’t mind and have time to check my next sentences, I expressed thousands of thanks to you..

Another issue that had to be addressed is the function evaluations consumed by these two methods. The X method solved all the problems at much lower cost than Y method and this superiority is most apparent at finer tolerance. This result is evidently shown the drawback of using reduction approach. Since X method required less total number of steps and function evaluation, then this method absolutely will have less computing time compared to Y method. Moreover the X method is more efficient as the tolerance becomes smaller.

On the comparison of the V, W and X will be discussed in terms of four parameters namely total steps taken, accuracy, function evaluations and execution time. This comparison is focuses at Problems 2.1 to 2.5. From analysis Tables 45-65, the total number of steps required by X method were cheapest followed by W method and then V method except for Problem 2.5. This achievement is expected since X method computed the solution at seven points simultaneously whilst W and V methods simultaneously computed the solutions at six and five points respectively. In other word it can be said that the higher order method will have larger step size then need lesser total number of steps.

In most cases X method is outperformed W and V methods. However, in some cases at certain tolerance W method have better or comparable accuracy with X method; for example in Problems 2.3, 2.4 and 2.5. Since V, W and X methods have the order of the method with five, seven and nine respectively then it is anticipated that the higher order method will give higher accuracy rather than the lower one.

It is notice that, the V method utilized fewest functions evaluation followed by W method and much computational cost taken by X method. This result is expected since V method utilized five functions per iteration whereas W and X methods utilized six and seven functions per iteration respectively. On the contrary, X method have least value of functions evaluation in Problem 2.3 at tolerance 0.003 and Problem 2.5 at tolerance 0.005. This might be happened due to the greatly reduction in the number of steps required. The computational time is fastest for V method followed by W and X methods even though it has more total steps. So that, the value of evaluate the called function is directly influence the computational time.

Next, Problems 3.1 to 3.4 that have been solved using W and X methods will be discussed numerically. From the overall performances, it can be seen that X method has superiority in terms of total number of steps and the value of maximum error. The total function calls is more expensive for X method except in Problems 3.2 to 3.4 for larger tolerance. This is due to the greatly reduction of steps required. For time consuming, W method is consistently efficient/cheapest for all prescribed tolerance.

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