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### Is quadrature still considered part of numerical analysis?

This question may admittedly sound strange, but having received several desk-rejects (all of them being based on being "out of scope" for the journal in question) from numerical analysis ...
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### Gaussian quadrature, with no exact result over polynomial, but on inverse functions

Generally, a Gaussian quadrature of degree $n$ over an interval $I$ is defined so that it integrates exactly polynomials up to degree $2n - 1$. The main tool are the orthogonal polynomials. When $I$ ...
I am looking for book recommendations or hints on numerical integration over infinite intervals. I am particularly interested in integrals of the form $\int\limits_{-\infty}^{+\infty} g(x) \exp(p_d(x))... • 131 4 votes 0 answers 266 views ### For noisy or fine-structured data, are there better quadratures than the midpoint rule? Only the first two sections of this long question are essential. The others are just for illustration. Background Advanced quadratures such as higher-degree composite Newton–Cotes, Gauß–Legendre, ... • 202 1 vote 1 answer 104 views ### Proof Reference - Polynomial interpolation at quadrature points If$\left( p_n \right)_{n=0}^{\infty}$is a family of orthogonal polynoamials with respect to a measure$\mu$on$[-1,1]$, and$\left( x_j, w_j \right)$are the quadrature points and weights for the ... • 3,554 0 votes 1 answer 155 views ### Clenshaw-Curtis integration without Fourier The Clenshaw-Curtis quadrature rule approximates an integral$I=\int\limits_{-1}^{1} f(x) \, dx$by $$I\approx I_n = \sum\limits_{j=1}^N f(x_j)w_j \, ,$$ where the$x_j$'s are the roots of the$N$-th ... • 3,554 4 votes 0 answers 288 views ### When (if ever) are the weights from Smolyak (sparse grid) cubature positive? Are there any$1$-dimensional quadrature rules of arbitrary accuracy, on either$[0,1]$or$\mathbb{R}$, with any non-trivial weight function, such that the associated$N$-dimensional cubature rule ... • 183 1 vote 2 answers 267 views ### Numerical Computation of Orthogonal Polynomials Recurrence Relations Background and notations: Given an interval$I\subseteq \mathbb{R}$and a continuous finite measure$d\mu = w(x)dx$, and denote$p_n(x)$the orthogonal polynomials with respect to$d\mu$. We have the ... • 3,554 9 votes 2 answers 1k views ### Recurrence of Legendre polynomial roots/ quadrature points Consider Legendre polynomials$p_n (x)$on$[-1,1]$. For each$n \in \mathbb{N}$we denote the zeros of$p_n (x)$by$\left( x_j ^{(n)} \right) _{j=1} ^n$. We know that these roots are distinct, and ... • 3,554 1 vote 1 answer 493 views ### PDF and CDF using Gauss-Legendre quadrature Consider the unit interval$I$with a continuous probability measure$\mu$, and consider a smooth random variable$f:I\to \mathbb{R}$. We can define its cumulative distribution function and ... • 3,554 6 votes 2 answers 762 views ### 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 ... • 3,554 4 votes 1 answer 792 views ### 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, ... • 3,554 2 votes 1 answer 161 views ### Does Gaussian Quadrature actually refer to Gauss-Legendre Quadrature？ 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 ... • 133 14 votes 2 answers 1k 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 abscissas and weights$\{ x_j, w^j \} _{j=1}^N$for large$N\in\mathbb{N}$. My question is how to do it,... • 3,554 4 votes 3 answers 1k views ### Quadrature formula max accuracy 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 ... • 43 2 votes 0 answers 162 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 ... • 121 2 votes 0 answers 1k 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 ... 16 votes 5 answers 3k 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) to ... 6 votes 1 answer 790 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) =...
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 ...