Quadrature refers to techniques in numerical integration, such as Riemann sum approximations, Simpson's rule and Gaussian quadrature.
Questions tagged [quadrature]
201 questions
6
votes
1 answer
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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5
votes
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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, and…
Collapse
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5
votes
2 answers
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
- 59
4
votes
2 answers
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 degree…
Siro
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4
votes
1 answer
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…
Physics
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4
votes
2 answers
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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4
votes
1 answer
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 $\sum$,…
J.G.
- 118,053
4
votes
0 answers
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 by…
Taco de Wolff
- 141
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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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3
votes
1 answer
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
- 10,162
3
votes
0 answers
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
- 717
3
votes
0 answers
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
- 1,016
3
votes
1 answer
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 first…
Il Prete Rosso
- 367
3
votes
1 answer
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
- 4,060
3
votes
2 answers
Weighting for Gauss-Legendre Quadrature
The textbook I am reading shows that the weighting of Gauss-Legendre Quadrature is
\begin{align*}
w(x_i) = \frac{1}{P_n'(x_i)}\int_{-1}^1 \frac{P_n(x)}{x-x_i} dx
\end{align*}
which is evaluated without proof to be
\begin{align*}
w(x_i) =…
Benjamin_Gal
- 306