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### Symmetric matrix formula for Gauss-Legendre quadrature

While searching the web, I came across the following algorithm for the Gauss-Legendre quadrature. I wasn't able to find a reference or a proof of my own as to why it works. I'll present it, and the ...
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### Computing Gauss Legendre Quadrature for Large N

I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscissas and weights $\{ x_j, w^j \} _{j=1}^N$ for large $N\in\mathbb{N}$. My question is how to do it,...
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I'm looking for a maximum accuracy quadrature formula: $$\int_{-1} ^{1} \sqrt{\frac {1-x}{1+x}} f(x)dx = A_1f(x_1)+A_2f(x_2)+R(f)$$ I don't know exactly if it's Trapezoidal rule which has the ...
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### Reference for the exponential decay of Legendre coefficients

In Short: I look for a reference to the proof that the spectral coefficients in the Legendre (or Jacobi) expansion are of exponential decay rate. Longer: If $p_n$ is the $n$-th Legendre polynomial, ...
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When the term Gaussian Quadrature appears in most Literatures, does it actually refer to Gauss-Legendre Quadrature. In other words, do they implicitly admit that they use the Legendre orthogonal ...
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### Area Under Generalized Parabolas and Hyperbolas without Calculus

This is shorter and more specific version of certain questions about a rather simple quadrature method. The answers I got were great but not what I asked. The terms in the title for $y=x^p$ look ...
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### Integrating Powers without much Calculus

I'll jump into the question and then back off into qualifications and context Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) ...
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### 2d quadrature weights for an arbitrary set of nodes

I need to estimate the value of a 2d integral $\int_{y_{min}}^{y_{max}}dy \int_{x_{min}}^{x_{max}} dx \, f(x,y) P(x,y)$ I have the an explicit analytical form for $P(x,y)$. I have samples of the ...
Gaussian quadrature allows us to integrate polynomials up to order $2 n-1$ using only $n$ function values. $\int_{x_0}^{x_1} ( \sum_{i=0}^{2 n-1} a_i x^i ) dx = f(a_0, \dots , a_{2 n-1})$ thus, the ...
I suspect this is a homework question somewhere, but I've not seen it elsewhere and it seems like it should be easy: let $f(x)$ be a concave function from $[0,1]$ to the reals such that \$f(0) = f(1) =...