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The following questions are from the 2018 professional course test of numerical analysis of 武汉岩石所:

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For the integral formula $\int_{0}^{h} f(x) d x \approx \frac{h}{2}[f(0)+f(h)]+\alpha h^{2}\left[f^{\prime}(0)-f^{\prime}(h)\right]$:

1. We need to determine the value of $\alpha$ to make it have the highest algebraic accuracy. 2. Suppose that the integrand f(x) has the fourth order continuous derivative, it needs to be verified that when $\alpha=\frac{1}{12}$, the integral remainder of this formula is $R(f)=\frac{1}{720} h^{5} f^{(4)}(\eta), \quad \eta \in[0, h]$.

Outer[Construct, {h/2 &, h/2 #1 &, h/2 #1^2 &, h/2 #1^3 &}, {0, 
     h}].{1, 1} + 
  Outer[Construct, {0 &, α*h^2 &, α*
       h^2 2 #1 &, α*h^2 3 #1^2 &}, {0, h}].{1, -1} == 
 Integrate[{1, x, x^2, x^3}, {x, 0, h}]

From the results of the above code, I can know that when $\alpha=\frac{1}{12}$, the integral formula $\int_{0}^{h} f(x) d x \approx \frac{h}{2}[f(0)+f(h)]+\alpha h^{2}\left[f^{\prime}(0)-f^{\prime}(h)\right]$ has the highest algebraic accuracy. But for the second question, I don't know how to verify it with MMA. I want to get as many methods as possible to verify the second problem.

Supplementary questions:

The following questions are from the 2015 professional course test of numerical analysis of 武汉岩石所:

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The main idea of this problem is to determine the coefficients A0, A1, A2, so that the integral formula $\int_{0}^{2 h} f(x) d x \approx A_{0} f(0)+A_{1} f(2 h)+A_{2} f^{\prime}(2 h)$ has the highest algebraic accuracy and gives the expression of the integral remainder.

Solve[((Outer[
       Construct, {#[0] &, #[2 h] &, #'[2 h] &}, {1 &, # &, #^2 &}] //
       Transpose).{A0, A1, A2}) ==
  Integrate[{1, x, x^2}, {x, 0, 2 h}], {A0, A1, A2}]

After comparison with the reference answer, we can know that the result of the above code is correct.

But when I use the Ulrich Neumann's method to solve this problem, I can't get the desired result:

Collect[Series[
  Integrate[
    f[x], {x, 0, 2 h}] - (A0*f[0] + A1*f[2 h] + A2*f'[2 h]), {h, 0, 
   5}], {h, h^_}]

I want to know how to use series expansion method to solve this problem.

The following questions are from the 2016 professional course test of numerical analysis of 武汉岩石所:

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I can't even know how to use series to expand $\int_{0}^{\infty} e^{-x} f(x) d x - A_{1} f\left(x_{1}\right)-A_{2} f\left(x_{2}\right)$. What should I do?

The following truncation error expression of Simpson's integral formula comes from this book:

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Integrate[
 Derivative[4][f][η]/
   4! (x - a) (x - (2 a + b)/3) (x - (a + 2 b)/3) (x - b), {x, a, b}]
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    $\begingroup$ I would expect \[Alpha]==-1/12 ! $\endgroup$ – Ulrich Neumann Oct 14 '20 at 10:29
  • $\begingroup$ @UlrichNeumann Thank you for your comments. Maybe I copied the question incorrectly. I hope you can provide the detailed code for my reference. $\endgroup$ – A little mouse on the pampas Oct 14 '20 at 10:33
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Answer to your first question:

Series[Integrate[f[x], {x, 0,h}] - (h/2 (f[0] + f[h]) + \[Alpha] h^2 (f'[0] - f'[h])), {h, 0,3}]
(* f''[0](\[Alpha]-1/12)h^3+O[h^4]*)

Optimal \[Alpha]==1/12

The answer to your second question follows from

Series[Integrate[f[x], {x, 0, h}] - (h/2 (f[0] + f[h]) +1/12 h^2 (f'[0] - f'[h])), {h, 0, 5}]
(*1/720 f''''[0] h^5*)     
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  • $\begingroup$ Your answer is great. Thank you. $\endgroup$ – A little mouse on the pampas Oct 14 '20 at 15:54
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    $\begingroup$ You're welcome! $\endgroup$ – Ulrich Neumann Oct 14 '20 at 15:57
  • $\begingroup$ I added a similar problem, but I don't know how to use the series expansion method to solve this problem, please help to solve it. $\endgroup$ – A little mouse on the pampas Oct 15 '20 at 0:49
  • $\begingroup$ Another difference is that the answer is $f^{(4)}(\eta)$ and yours is $f^{(4)}(0)$. $\endgroup$ – A little mouse on the pampas Oct 15 '20 at 2:17

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