# Questions tagged [quadrature]

Quadrature refers to techniques in numerical integration, such as Riemann sum approximations, Simpson's rule and Gaussian quadrature.

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### When can one trust that an adaptive quadrature scheme is not missing nontrivial areas of the domain?

hcubature http://ab-initio.mit.edu/wiki/index.php/Cubature_(Multi-dimensional_integration) seems to be a useful package for computing multidimensional integrals numerically. However, I have come upon ...
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### How to interpret/understand polynomial r in proof for the weights in the Gauss rule being positive

I am trying to proof the following statement: "the weights in the Gauss rule are positive:" by definition $w_i = \int_{a}^{b}Li$ ( from now on notated as I[Li] ) with Li being the Lagrange ...
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According to Chapter 5 of Numerical Methods and Software by Kahaner, et al. (1989), it can be shown that the error associated with Gaussian quadrature is $\displaystyle\int_a^b f(x)\,dx - \sum_{i=1}^... 0answers 22 views ### Quadrature of rather symmetric moderate-dimensional functions I'm interested in doing some quadrature in dimension 5, .., 10 so am thinking of sums like $$\sum_{1 \leq i, j, \ldots k \leq n} f_{i, j,\ldots, k}$$ the summands corresponding to the contribution ... 0answers 48 views ### Can a recursive, continuous integral be approximated with Gauss-Legendre or similar? Maybe I need to reformulate(?). Suppose there is this simple function: $$f(x)=\int_a^b{x \text{d}x}$$ If it were to be discretized, there would be losses due to sampling. In order for the errors to ... 1answer 182 views ### Gauss-Legendre vs Gauss-Chebyshev quadratures (and Clenshaw-Curtis) There's a point I wish to elucidate about integrating polynomials. It is known that functions are better approximated by polynomials with non-uniformly spaced nodes, which is why Gauss quadratures ... 0answers 177 views ### how to implement adaptive gaussian (kronrod?) quadrature (technicalities) I am trying to figure out what is the best way to implement an adaptive quadrature scheme which preferentially makes use of guassian quadrature. Guassian quadrature doesn't really give any easily ... 1answer 159 views ### Ancient Greek proofs of Archimedes' three properties of the parabola? Please refer to the document, "Archimedes' Quadrature of the Parabola": https://www2.bc.edu/mark-reeder/1103quadparab.pdf This document describes how Archimedes proves that the area of any parabolic ... 2answers 267 views ### High accuracy root finder of Legendre polynomials' derivatives? I need GLL (Gauss-Legendre-Lobatto) nodes for the Legendre-Galerkin-NI spectral method. It requires me to find the roots of the derivatives of Legendre polynomials. My Matlab program calculates the ... 1answer 122 views ### Why not simply use sine weights with Clenshaw-Curtis nodes? Clenshaw-Curtis quadrature is based on writing $$\int_{-1}^{1} f(x)dx=\int_{0}^{\pi}f(\cos y)\sin y dy$$ and then replacing$f(\cos y)$by a truncated Fourier series, so that the integral can be ... 1answer 99 views ### the highest degree of the polynomial, for which the above formula is exact? Consider the quadrature formula$\int _{-1}^1\vert x \vert f(x)dx \approx \frac{1}{2}[f(x_0)+f(x_1)]$, where$x_0$and$x_1$are quadrature points. Then the highest degree of the polynomial, for which ... 1answer 183 views ### Interleaving of Gaussian quadrature nodes and weights A Gaussian quadrature is used to approximate the following integral: $$\int_{-1}^{1} f(x) dx \approx \sum_{i=1}^n w_i f(x_i).$$ Numerically I've found an interesting property of$x_i$and$w_i$: if ... 1answer 49 views ### Finding formula for error in the basic midpoint rule I wanted to derive the formula for the error in the basic midpoint rule. For the error I found $$E(f)= \int_{a}^{b} f[\tfrac{a+b}{2},x](x-\tfrac{a+b}{2})\,dx.$$ I didn't know how to go from here so ... 2answers 113 views ### How to obtain the coefficients of a trigonometric polynomial by evaluating at well-chosen points? Suppose that we are given the trigonometric polynomial $$P_M(t)=a_0 +\sum_{m=1}^M a_m \cos(mt)+b_m\sin(mt),\quad t\in[-\pi, \pi].$$ Question. How to find nodes$t_1, t_2\ldots t_K\in [-\pi, \pi]...
Consider the following quadrature: $$\int_0^1 x^cf(x)dx\approx Af(0)+B\int_0^1 f(x)dx, c>1, \neq 0$$ Determine A and B such that this rule has a degree of exactness 1. Let $E(f)$ be the error ...