A New First Order Expansion Formula with a Reduced Remainder

Round 1

Reviewer 1 Report (New Reviewer)

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Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report (New Reviewer)

The paper is good, but some revision is needed. See the attached file.

I congratulate the authors for their ideas and for their results.

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report (Previous Reviewer 2)

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Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report (New Reviewer)

No comments

Author Response

No comment

Reviewer 3 Report (Previous Reviewer 2)

The authors are not including properly all comments and suggestions. They must include the following important suggestions in the manuscript:

Page 4, in the proof of Lemma 2.3: Provide detail for n=2 before using induction for n. The authors comment is not suitable for justification "Other particular values of n is unnecessary from a mathematical point of view, and then, it will artificially overweight the manuscript."

Generally, when we use induction, we try to provide at least three terms in computation. For n=2, it is not unnecessary.

Page 8: Formula (33) is the direct application of formula (7) when n=2. It should justify using numerical values of three points.

Here, the authors should choose functions as well as numerical values of a and b. Then justify formula (7).

 

Author Response

Dear reviewer,

As we wrote you last time, the very well-known induction procedure does not require to initialize the concerned statement S(n) more than one given value of the integer n.

Thousand books and papers repeat exactly the same:

Let S(n) be a statement involving n. If
(i) S(1) holds, and
(ii) for every k ≥ 1, S(k) → S(k + 1),
then for every n ≥ 1, the statement S(n) holds. 

See for example, 

1. Peano, G. (1889). Arithmetices Principia nova methodo exposita. Bocca : Torino.
2. Handbook of Mathematical Induction: Theory and Applications of  David S. Gunderson, 2010, (Chap 3, Page 35),
3. https://en.wikipedia.org/wiki/Mathematical_induction,
4. and so on, ...

The second point you mentioned has been taken into account and we gave an example (see page 8).

However, we do not understand when you write "Then justify formula (7)", since we did explain the motivations which lead us to derive formula (7). 

Prof. Joel Chaskalovic

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

A new Taylor type first order formula using a suitable linear combination of the first derivative of the function is stated. This new formula allows to minimize the remainder term of the classical Taylor formulas. It is also showed how to chose the weights of the linear combination in order to minimize the new remainder.  

The new Taylor formula is applied to the estimate  the Lagrange interpolation error by means of polynomial of degree 1 and to the trapezoidal quadrature rule.

It is shown that for the particular case of the periodic function then the error is halved with respect to the classical one.

The proposed formula seems to be new. The proofs are correct. Anyway it seems to me that the results are more theoretical than really practical. In other words from the numerical point of view use this kind of new estimates do not really improve the approximation process. More interesting a possible application in using the values of the derivative of the function when available.

 

Author Response

Dear reviewer,

Thanks for your approbation.

Best regards,

Prof. Joel Chaskalovic

Reviewer 2 Report

Please see attached file

Comments for author File: Comments.pdf

Author Response

Dear reviewer,

Thanks for remarks. Please find here my answers:

• In Page 2, we define separately the function Φ before Proposition 2.1
• Done.
• As well known, the first step of mathematical induction only requires to verifiy that the concerning property (Pn) is true for a given initial value of n; More is overqualified and a lost of time.
• Done.
• Done.
• Formula (34) is simply the direct application of formula (8) when n=2.
• Done.
• Numerical results will be investigated in a future paper. Here we limit ourselves to derive theoretical analysis. 
• Done.
• Done.
• Since this work is an extension of previous works, it seems obvious that self-citations appear.

Sincerely,

Prof. Joel Chaskalovic

Reviewer 3 Report

 

Title: “A new first order Taylor-like theorem with an optimized reduced remainder”

This manuscript is devoted to provide a new first order Taylor-like formula where the authors claim that the corresponding remainder is reduce  in comparison with the usual one which appears in the classical Taylor’s formula. The authors, to derive this new formula, introduced a linear combination of the first derivative of the concerned function, which is computed at n + 1 equally-spaced points between the two points where the function has to be evaluated. We show that an optimal choice of the weights in the linear combination leads to minimizing the corresponding remainder. Then, the authors analyzed the Lagrange P1- interpolation error estimate and also the trapezoidal quadrature error, in order to assess the gain of accuracy we obtain using this new Taylor-like formula.

 

General comment: Although the topic of this work is interesting, if I read the manuscript, I cannot see the value of this work at this stage.

Indeed, the authors simply added a linear combination of the first derivative of a given function calculated in some chosen points. It seems to be quite trivial that the remainder is lower than in the classic fomula which does not have any addition. Thus, to have a reduced remainder can not be the only main aim of this work. Moreover, the authors should explain in detail why a reader should use a first order approximation series when a second order approximation is possible. The authors should find a better way to convey their message. In addition, they should provide some more examples of a possible use of their formula to better approximate the Taylor expansion of functions. Please rework the main text accordingly, better underlining what are the main results and quantifing the value of their work through numbers.

Author Response

Dear reviewer,

Thank you for your remarks and, please find our answers below:

To our opinion, the result of Theorem 2.2 is not trivial, since before getting the proof, it is not obvious that the related weights we found out minimize the corresponding remainder for a set of equally space points xi.

Then, the only main of this work is not to find a "reduced remainder", but the smallest one.

Furthermore, based on Theorem 2.2, we highlight the first consequences we derived regarding the improvement of the bounds, either for the interpolation estimate, or for the approximation estimate in the context of finite elements, or for the trapezoidal quadrature.

From our point of view, it sufficiently important for further applications. It is the reason why, we mention that these results concern functions which only belong to C²([a,b]). As a consequence, availability of second order formula is not valid anymore, (see for example remark 3, page 12).

Sincerely,

Prof. Joel Chaskalovic

Reviewer 4 Report

This paper presents a supposedly new first-order Taylor-type formula. Actually it is not new, formula (8) can be obtained from the Composite Trapezoidal Rule applied to \int_a^b f'(x)dx taking n subintervals.

Note also that the Fundamental Theorem of Integral Calculus mentioned in Remark 2 is not so. This is not the fundamental theorem of integral calculus, it is simply an identity obtained after the change of variable x=a+(b-a)t in the integral \int_a^b f'((x) dx.

Similarly, the formula in (34) results from the application of the Composite Trapezoidal Rule with n=2 to the integral \int_a^b f'(x) dx.

On the other hand, I disagree that the new reminder is 2n times smaller than the reminder of the Taylor formula. The truth is that the bound of the absolute value of the new remainder is 2n times smaller than the bound of the absolute value of the remainder of the Taylor formula.

The fact of having found a finer bound may be of some interest, but not to write an article like the one presented. I do not find the paper with merits enough to be published.

Author Response

Dear reviewer, 

Thanks for your remarks and please, find our answers below:

• Indeed, Formula (8), and then (34), can be interpreted as an application of the composite trapezoidal rule to function f' integrated on [a,b]. However, these formula do  not appear in the litterature as a generalized formula of the classical Taylor's one, and especially not with the aim to get a smaller bound for the new remainder. Anyway, it is an interesting remark and we will include it in the manuscript.
• We changed the terminlogy of "fundamental theorem of integral calculus". But, for your knowledge it is used by several authors, (see for example, A. Avez, Calcul différentiel, Ed. Masson).
• It an usual abuse of terms in the context of estimates. Anyway, we precised that it concerns the bounds.
•  From our point of view, improvements of the bounds of estimates which appears in numerical analysis and their consequences, justify the way we write this paper based on formula (8).  

Sincerely,

Prof. Joel Chaskalovic

Round 2

Reviewer 3 Report

 

Title: “A new first order Taylor-like theorem with an optimized reduced remainder”

This manuscript is devoted to provide a new first order Taylor-like formula where the authors claim that the corresponding remainder is reduce  in comparison with the usual one which appears in the classical Taylor’s formula. The authors, to derive this new formula, introduced a linear combination of the first derivative of the concerned function, which is computed at n + 1 equally-spaced points between the two points where the function has to be evaluated. We show that an optimal choice of the weights in the linear combination leads to minimizing the corresponding remainder. Then, the authors analyzed the Lagrange P1- interpolation error estimate and also the trapezoidal quadrature error, in order to assess the gain of accuracy we obtain using this new Taylor-like formula.

General comment: Unfortunately the authors did not revise the manuscript. As a consequence, some big problems remain to be solved in this work. Some of them are listed below:

1) The authors claim that they provide “ A new first order Taylor-like theorem with an optimized reduced remainder”. However, the “philosophy” of the Taylor formula seems to be quite different. Indeed, in Taylor series only an expansion around a single point is used. The authors used this formula with the addition of further derivatives calculated in chosen points in the definition interval. As a consequence, it is trivial that the remainder is lower that the “classic” Taylor formula. It could be interesting to understand whether this is really a “Taylor like” formula or something of different.

2) With reference to the point 1): The authors should show that their formula in Eq (8) is really novel. Indeed, it could seem quite similar to an application of other well known formulas (e.g., the composite trapezoidal rule, etc.), which already use the combination of values of the function in some points of a given interval. Are the authors able to show this ?

3) The main theorem of this work is not general. Indeed, it is not clear why they only consider functions \in C^2[a,b]. What about a generalization of this theorem ? Could the interested readers use a generalization of Eq (8) for high degree of derivatives ? Could they have a smaller (reduced) remainder with Eq (8) with derivative of order n with respect of the classic Taylor formula with derivative of order n+1 ? Is it valid for every integer value of n ? Please use a theorem to show this, then the presented results will be only a consequence for n=2.

 

Author Response

 "Please see the attachment."

Author Response File: Author Response.pdf

Reviewer 4 Report

I have read carefully the paper, and do not find merit enough to be published. I still maintain that the formula in (8) can be obtained after the application of the composite trapezoidal rule. 

I do not agree even with the title. In view of what the formula in (8) can be named as a Taylor-like expansion? Taylor's formula is a local formula about a given point x_0 which uses higher order derivatives at x_0. But in formula (8) there appears only the first derivative evaluated at different points on [a,b]

Concerning the Fundamental Theorem of Calculus mentioned in a previous version, it is true that it appears in Calcul Différentiel (Ed. Masson) by A. Avez, but I'm that the authors agree that this is not the usual denomination. It's the first time I see this.

The statement of Theorem 2.2 is badly formulated.  If the weights ω_k(n), (k = 0, n), satisfy that the sum equals 1 then we have that ω_0(n)=ω_n(n)=1/(2n), ω_k(n)=1/n for 0<k<n. Those values give one possible solution, but there are infinite of them. The sum of the weights ω_k(n) equal to 1 is an hypothesis of Theorem 2.2, but in the proof it appears that this is an assumption (Let us assume for the sake of simplicity (see Remark 3 below) that \sum_{k=0}^n ωk(n) = 1). This is confusing, even considering the comments in Remark 3.

After (31) it appears "by summing on k between 0 and n", which is wrong. In fact it should be from 0 to n-1.

It is not clear the final identity in (33), nor why the values of the weights are those in the following line after (33). 

In (43) it should read f''(\xi_x) instead of  f''(\xi).

It is not clear how is obtained the improved interpolation polynomial in (57). How is this polynomial obtained? What does "improved" mean?

In (61) it is h=b-a. What is the value of h*?

It is not clear the first conclusion in page 15 since your are comparing bounds for different formulas, that is, for the trapezoidal rule and the corrected trapezoidal rule. This is not fear.

On the other hand, from formula (77) assuming that f'(a)=f'(b), the second conclusion is trivial.

Finally, since Cheng and J. Sun proved in [16] which is the best constant in a similar bound as in (77), the constant 1/48 provided in (77) is of little value.

 

Author Response

"Please see the attachment."

Author Response File: Author Response.pdf