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I'm struggling with the following problem:

What is the maximum degree of exactness that we can obtain with the following quadrature >formula $$\int_0^1 f(x)\frac{1}{\sqrt{x}}dx \approx w_0 f(x_0) + w_1 f(x_1)$$

Compute weights and nodes

I should use some theorem, but I can't understand which one! Also, I think that the maximum degree is $r=3$, because in this case, imposing the exactness I will end up with a system of $4$ equations in $4$ unknowns.

I choose a basis of $\mathbb{P}^{3} = \{1,x,x^2,x^3\}$, and I obtain

\begin{cases} w_0+w_1 = 2 \\ w_0x_0 + w_1x_1 = \frac23 \\ w_0x_0^2 + w_1x_1^2 = \frac25 \\ w_0 x_0^3 + w_1 x_1^3 = \frac27 \\ \end{cases}

but the solution seems too hard to do by hand. Am I missing something? How can I decide a priori the degree of exactness.

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You need to use the so called Gauss–Jacobi quadrature with $\alpha = 0$ and $\beta = -\frac{1}{2}$. If you look at the error term, you see that you can integrate exactly for a degree up to $2n-1$ with a $n$ points scheme.

EDIT to truly answer your question.

You want to use this theorem about Gaussian quadrature, theorem 3.6.12 in [1] for a book reference. In your case for two points with the weight function $x \mapsto \frac{1}{\sqrt{x}}$ on the interval $(0,1)$, you want to find the roots of $P_2$ where $(P_n)_{n \ge 0}$ are the orthogonal polynomials associated with the scalar product $$ \langle f,g\rangle = \int_0^1 f(x)\, g(x)\, \frac{\mathrm{d}x}{\sqrt{x}}. $$

One way to do it is:

  • Set $P_0 = 1$.
  • Find $P_1 = x + a$ such that $\langle P_0, P_1\rangle = 0$.
  • Find $P_2 = x^2 + bx + c$ such that $\langle P_0, P_2\rangle = 0$ and $\langle P_1, P_2\rangle = 0$.
  • Find the roots of $P_2$ for the points $x_0$ and $x_1$.
  • Solve for example $$ \begin{cases} w_0 + w_1 = 2\\ w_0 x_0 + w_1 x_1 = \frac{2}{3} \end{cases} $$ to get the weights $(w_0, w_1)$.

[1] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer, 2002.

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  • $\begingroup$ So I can integrate exactly polynomials of degree $3$ , as I expected. But in any case, I don't know how to compute the weights and nodes by hand, because this exercise was in a written exam $\endgroup$ – lukk Sep 9 at 11:22
  • $\begingroup$ I have edit my answer to add details. $\endgroup$ – Zoïs Moitier Sep 9 at 15:31

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