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In this section, the accuracy of the ABC method is analyzed by examining the performances of seven test problems as used/tested earlier in Chapter 2. The results obtained were compared with the method in Richard (2006) such as shown in Tables 56-60. The same notations explained in section 2.6 were used in this section with the new notations given as

Which one is correct to use in this sentence? Used or tested? Thanks a lot..
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In this section, the accuracy of the X method is analyzed by examining the performances of seven test problems -

from Chapter 2.

previously tested in Chapter 2.

already used in Chapter 2.

seen in Chapter 2.

In the current instance - I presume you are analyzing the accuracy of results from seven problems that were solved/tested/used and published in Chapter 2.

If this is the case, any of the above would work.
Thanks a lot JP..How about these sentences? Is it suitable to use ‘less’ in these sentences? And do the bolded sentence sounds complex?

Since the X method required less total number of steps and function evaluation, then this method absolutely will have less computing time than the Y method. In addition, as the tolerance becomes smaller, the X method is getting more efficient since the X method is able to reduce the execution time more than half than consumed by Y method.
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What do you think of the following?

Since method X has shown that it will require fewer steps and function evaluations, it will certainly require less computing time than method Y.

As tolerance is reduced, method X increases efficiency because it lowers execution time by more than half (or -by more than 50%) of the time required by method Y.
Yes exactly..

Another question to ask are:

The X method requires fewest steps followed by W and V methods. How to describe that? Does this sentence correct? Thanks a lot..

"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."

Does this sentence have any grammatical error? Thanks a lot..

"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."
Another question to ask are:

The X method requires fewest steps followed by W and V methods. How to describe that? Does this sentence correct? Thanks a lot..

This sentence is not correct - I need more information to answer your questions.

"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."

Does this sentence have any grammatical error? Thanks a lot..

"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."
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This is the full text.. Thanks a lot JP.

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 Table 46. 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/requires 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, six and seven respectively then it is anticipated that the higher order method will give higher accuracy rather than the lower one.
Myvi - I am not thrilled with my last paragraph (in blue) but I think it captures the essentials. I would need more detail from you to be completely satisfied. If you need additional help, please contact me directly using my email (found on my profile).

Tables 45-65 (with the exception of Table 46) show that method X required the least number of steps compared with methods W and V. This was to be expected, as method X simultaneously computed the solution at seven points while methods W and V simultaneously computed the solution at six and five points, respectively. In other words, even though the higher order method will have larger step sizes than needed, it will require using fewer total steps.

In most cases method X outperformed methods W and V. However, at certain tolerance levels method W showed better or comparable accuracy compared with method X. For example, in Problems 2.3, 2.4 and 2.5. methods V, W and X show orders of method of five, six and seven, respectively. The higher order method can be expected to have higher accuracy compared with the lower ones.
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