Mathematics

Questions tagged [quadrature]

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

203 questions
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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 we split $[-1,1]$ interval into subintervals $I_i…
uranix
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Gauss Legendre quadrature problem with Legendre polynomials composed with square root

Let $P_n$ be the orthogonal Legendre polynomial with a degree of $n$ , meaning it satisfies the following recursive formula: $$(n+1)P_{n+1}(x)-(2n+1)xP_n(x)+nP_{n-1}(x)=0$$ where $P_0(x) = 1$ and $P_1(x) = x$ . Let $w(x)=1$ be the weight function,…
Collapse
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Cubic Bézier curve arc length parametrization reversal: find t given a length

I am following this paper Approximate Arc Length Parametrization, M. Walter & A. Fournier, 1996 and have succesfully implemented the direct solution, as in finding the length $s(t)$ given $t$ . This algorithm calculates the length of a curve given…
Taco de Wolff
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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 segment is equal to four-thirds the area of the…
TomF
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First derivative of Legendre Polynomial

Legendre polynomials ( $P_n$ ) are defined as a particular solution to the ODE. $$(1-x^2)P_n^{''}-2xP_n^{'}+n(n+1)P_n=0$$ It is expressed by Rodrigues’ formula. $$P_n=\frac{1}{2^nn!}\frac{d^n}{dx^n}((x^2-1)^n)$$ I was given this formula for its…
Il Prete Rosso
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How to find points for 1-D Gauss-Legendre Quadrature for Double Integral

For single integrals, Gauss-Legendre quadratures approximate the integral, as follows: $$\int_{-1}^1 f(x)\text{ }dx\approx\sum_{j=1}^nw_if(x_i).$$ By choosing the correct $x_i$ , this approximation can be made to be exact for polynomials up to…
Siro
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FUN with f̶̶l̶̶a̶̶g̶̶s̶ Newton Cotes Quadrature formula and Bernoulli polynomials of the second kind

I was told to phrase my question in a more exciting way when I asked it last time. The following is a preliminary consideration. If you don't need it, just scroll down to START HERE. Here we go then: Imagine you have a really cool integral,…
Physics
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Why is the Gaussian-Legendre Quadrature so effective?

I understand how it works, how its derived, etc. The proof of it has been shown to me. That is to say, I know how Legendre polynomials are derived, I know they are orthogonal, I know we sample a function at the roots of the polynomial, I know how…
CogitoErgoCogitoSum
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What's the name for this numerical integration algorithm?

Suppose I estimate $\int_0^1 f(x)dx$ as $\frac12(f(a)+f(b))$ , with $a,\,b$ chosen to achieve the lowest-order possible error. We assume $f$ equals its Maclaurin series and $\int$ commutes with $\sum$ , viz.…
J.G.
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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 $-1 < x_1 < x_2 < \cdots < x_n < 1$, $x_{n-i+1} = -x_i$, $w_{n-i+1} =…
Calum Gilhooley
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When can the sum of Gaussian-shaped terms over integers be approximated by the corresponding Gaussian integral

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 $\{k\in \mathbb Z: -10n^{3/4} \leq k \leq O(1) -\frac{\sqrt{n}}{24}\}$ .…
tutoto
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Trapezoidal Rule on Infinitely Differentiable Periodic Functions

If I understand it correctly, the Euler-Maclaurin summation formula states that for a periodic and infinitely differentiable function, the error of the trapezoidal rule of the numerical integration of the function over an integer multiple of the…
velut luna
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Integration over ball

I have the following exercise: Evaluate the multivariate integral $$\int_Af(\boldsymbol x)\,\mathrm d\boldsymbol x$$ numerically, where $f:\mathbb R^p\to\mathbb R$ is a known function and $A := \{\boldsymbol x\in\mathbb R^p : \Vert \boldsymbol…
Syd Amerikaner
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Flaw in proof about an exact quadrature method

Consider the quadrature method $$ \int_{-1}^1f(x)dx \approx \sum_{k=0}^N w_kf(x_k), $$ where $x_0=-1, x_N=1$ , and $x_1,\ldots,x_{N-1}$ are the roots of the derivative of the degree- $N$ Legendre polynomial $P'_N$ . This is called Lobatto quadrature…
Frank Seidl
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power series, uknown substitution of $-1$

This time I type my problem into LaTeX, but to show the problem I'll also include the appropriate snippet: I do not understand in the book by Ralston: A first course in numerical analysis $(4.10-28)$ why $$\Sigma_{j=0}^m |c_{mj}|=t_m(-1):$$ I would…
user122424
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