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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1answer
230 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 ...
3
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0answers
51 views

The Distribution of Abscissae and Sums of Weights in Gaussian Quadrature

Let $n$ be a positive integer, and for $i = 1, 2, \ldots, n$, let $x_i$ be the $i^\text{th}$ abscissa for $n$-point Gaussian quadrature, and $w_i$ the associated weight, so that $-1 < x_1 < x_2 &...
2
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1answer
88 views

Gaussian quadrature: Orthogonal polynomial for chi distribution

I have a problem involving numerical integration of the form: $$I = \int_0^\infty \!dx \, w(x) f(x)$$ where the weighting function is a chi distribution of degree 2, i.e., $$w(x) = x \, e^{\frac{-...
1
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0answers
267 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 ...
1
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1answer
45 views

Deriva parameters of Gauss quadrature rule

Derive the parameters of Guass quadrature with three points $$ \int_{-1}^1 f(x)\,\mathrm dx \approx C_1 f(\xi_1)+C_2 f(\xi_2) + C_3 f(\xi_3) $$ such that the integral is exact up to $x^5$. ...
0
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2answers
162 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]...