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9
votes
2answers
238 views

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 abscisas and weights $\{ x_j, w^j \} _j$ for large N. My question is how to do it, and why should it ...
2
votes
0answers
39 views

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 ...
1
vote
0answers
614 views

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 ...
10
votes
4answers
1k views

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) ...
6
votes
1answer
378 views

Approximation of an integral of a concave function

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) ...
1
vote
1answer
304 views

gaussian quadrature

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 ...