Questions tagged [quadrature]

Question 1

tl;dr: Are there $t$ points $x_1,\dots,x_t \in \mathbb{R}$ such that $$ \frac{1}{t} \sum_{i=1}^t x_i^k = \int_{-\infty}^\infty x^k\ d\rho(x)$$ for every $1 \leq k \leq t$, where $d\rho$ is the ...

Question 2

I have the following finite sum: $$\frac{1}{\sqrt{n/12}}\sum_{k\in I} \exp\left(-\frac{(\frac{k}{\sqrt{n/12}} + \frac{1}{2\sqrt{3}})^2}{2}\right),$$ where$I$ is the interval $\{k\in \mathbb Z: -10n^{3/...

Question 3

I am trying to numerically integrate the following: $I=\int_0^{d_2} \int_0^{w_2} \int_0^{d_1} \int_0^{w_1} g(x_1,y_1,x_2,y_2) e^{(-\frac{1}{\sigma^2}((x_1-\alpha_1)^2+(y_1-\beta_1)^2+(x_2-\alpha_2)^2+(...

Question 4

Consider the integral $$ I_1 = \int_0^1 ds\, f(s). $$ This integral can be approximated using, e.g., a Gauss-Legendre quadrature rule. Let's call the result $I_1^{\rm GL}$. Now suppose that I want to ...

Question 5

Consider the definite integral $$ I = \int_{-1}^1 ds \, f(s). $$ Suppose that for some reason I insist to numerically evaluate this integral using an ODE solver. I could define $$ I(t) = \int_{-1}^t ...

Question 6

I want to numerically perform an integral of the form $$ I = \int_{-\infty}^{\infty} dx\, g(x) f(x), $$ where $g(x)$ is a complex-valued function of the real variable $x$ that only has nonnegligible ...

Question 7

I was trying to understand (at a really high level) how Gauss–Legendre quadrature works and, from what I understand, it allows to exactly compute the result of an integral for a given polynomial of ...

Question 8

Suppose $f\in C^k([0,1])$ where $C^k$ is the space of functions with continuous $k$-th derivatives. I wish to approximate the integral $I:= \int_{[0,1]} f(x) dx$ given the values of the function $(f(...

Question 9

Double exponential integration formulas were proposed by Takahasi and Mori. This technique is widely appreciated for exponential convergence (numerical integration error vs number of samples). However,...

Question 10

I am working on the numerical solution of a system of singular integral equations. I already discretized the system using a Gauss-Chebyshev quadrature and I thus obtain a linear system with the ...

Question 11

Given a random process $X(t)$, it's mean can be found as $$\langle X(t) \rangle = \lim_{T \rightarrow \infty} \frac{1}{T}\int_0^T X(t) \mathrm{d}t$$ The normal way of estimating this would be to ...

Question 12

Suppose I have a Brownian motion $(B_t)_{t \in [0,1]}$ and I want to numerically calculate the integral $\int_0^1 B_s \,\mathrm ds$. To do this, I first sample a single path of Brownian motion at ...

Question 13

What is the order of convergence of the Gauss quadratures? I.e. By what factor of the interval $h = b-a$ does the Gauss quadratures scale by? For the Simpson rule and Trapezoidal rule you get a nice ...

Question 14

I am interested in whether an exact cubature to the following problem is possible: I have a convex, bounded polytope $H$ over $t \in \mathbb{R}_+$ and $x \in \mathbb{R}^n$. I am interested in the ...

Question 15

I was doing some calculations and this sum popped up : $$S_n=\sum_{j=1}^{n}\frac{w_j}{(1+x_j)^2}$$ Where $w_j$ and $x_j$ are the Gauss-Legendre weights and points associated to $n$ that represents the ...

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Questions tagged [quadrature]

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Order of convergence of Gauss quadrature of n nodes

Exact cubature for rational polynomials over polytope with single dimensional quotient polynomial

Cool property about gauss-legendre quadrature

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