The following questions are from the 2018 professional course test of numerical analysis of 武汉岩石所:
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:
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:
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:
Integrate[
Derivative[4][f][η]/
4! (x - a) (x - (2 a + b)/3) (x - (a + 2 b)/3) (x - b), {x, a, b}]
\[Alpha]==-1/12! $\endgroup$