Questions tagged [quadrature]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1
vote
1answer
112 views

Integrating a B-Spline basis function with respect to the standard normal PDF

I am looking for ways to evaluate exactly (i.e. analytically or semi-analytically) integrals of the type: $$ \int_{-\infty}^{+\infty}B_{i}^k(u)e^{-\frac{(u-\mu)^2}{2\sigma^2}}du, $$ where $B_i^k$ is a ...
0
votes
0answers
13 views

Low order quadrature for low order terms in finite element method- convergence analysis

Consider the PDE $$\frac{\partial u}{\partial t} = \Delta u - \kappa u,$$ posed over a bounded domain $\Omega$ (say, $\Omega \subseteq \mathbb{R}^2$). Assume we have a family of triangulations $\...
1
vote
0answers
46 views

Explicit growth rate estimation of Gauss-Laguerre quadrature

The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0;+ \infty[$ by a finite sum, according to: $ \displaystyle { \int _0 ^{+ \infty} ...
2
votes
1answer
138 views

Error in Gauss-Laguerre numerical quadrature scheme

The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0 ; \infty[$ by a finite sum, according to: $$ \int _0 ^{+ \infty} ...
0
votes
1answer
137 views

Gaussian quadrature, with no exact result over polynomial, but on inverse functions

Generally, a Gaussian quadrature of degree $n$ over an interval $I$ is defined so that it integrates exactly polynomials up to degree $2n - 1$. The main tool are the orthogonal polynomials. When $I$ ...
2
votes
1answer
248 views

Quadrature for numerical integration over infinite intervals

I am looking for book recommendations or hints on numerical integration over infinite intervals. I am particularly interested in integrals of the form $\int\limits_{-\infty}^{+\infty} g(x) \exp(p_d(x))...
4
votes
0answers
190 views

For noisy or fine-structured data, are there better quadratures than the midpoint rule?

Only the first two sections of this long question are essential. The others are just for illustration. Background Advanced quadratures such as higher-degree composite Newton–Cotes, Gauß–Legendre, ...
1
vote
1answer
90 views

Proof Reference - Polynomial interpolation at quadrature points

If $\left( p_n \right)_{n=0}^{\infty}$ is a family of orthogonal polynoamials with respect to a measure $\mu$ on $[-1,1]$, and $\left( x_j, w_j \right)$ are the quadrature points and weights for the ...
0
votes
1answer
124 views

Clenshaw-Curtis integration without Fourier

The Clenshaw-Curtis quadrature rule approximates an integral $I=\int\limits_{-1}^{1} f(x) \, dx$ by $$I\approx I_n = \sum\limits_{j=1}^N f(x_j)w_j \, ,$$ where the $x_j$'s are the roots of the $N$-th ...
3
votes
0answers
205 views

When (if ever) are the weights from Smolyak (sparse grid) cubature positive?

Are there any $1$-dimensional quadrature rules of arbitrary accuracy, on either $[0,1]$ or $\mathbb{R}$, with any non-trivial weight function, such that the associated $N$-dimensional cubature rule ...
1
vote
2answers
238 views

Numerical Computation of Orthogonal Polynomials Recurrence Relations

Background and notations: Given an interval $I\subseteq \mathbb{R}$ and a continuous finite measure $d\mu = w(x)dx$, and denote $p_n(x)$ the orthogonal polynomials with respect to $d\mu$. We have the ...
6
votes
2answers
832 views

Recurrence of Legendre polynomial roots/ quadrature points

Consider Legendre polynomials $p_n (x)$ on $[-1,1]$. For each $n \in \mathbb{N}$ we denote the zeros of $p_n (x)$ by $\left( x_j ^{(n)} \right) _{j=1} ^n$. We know that these roots are distinct, and ...
1
vote
1answer
297 views

PDF and CDF using Gauss-Legendre quadrature

Consider the unit interval $I$ with a continuous probability measure $\mu$, and consider a smooth random variable $f:I\to \mathbb{R}$. We can define its cumulative distribution function and ...
6
votes
2answers
531 views

Symmetric matrix formula for Gauss-Legendre quadrature

While searching the web, I came across the following algorithm for the Gauss-Legendre quadrature. I wasn't able to find a reference or a proof of my own as to why it works. I'll present it, and the ...
4
votes
1answer
603 views

Reference for the exponential decay of Legendre coefficients

In Short: I look for a reference to the proof that the spectral coefficients in the Legendre (or Jacobi) expansion are of exponential decay rate. Longer: If $p_n$ is the $n$-th Legendre polynomial, ...
2
votes
1answer
131 views

Does Gaussian Quadrature actually refer to Gauss-Legendre Quadrature?

When the term Gaussian Quadrature appears in most Literatures, does it actually refer to Gauss-Legendre Quadrature. In other words, do they implicitly admit that they use the Legendre orthogonal ...
14
votes
2answers
960 views

Computing Gauss Legendre quadrature for large $N$

I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscissas and weights $\{ x_j, w^j \} _{j=1}^N$ for large $N\in\mathbb{N}$. My question is how to do it,...
4
votes
3answers
735 views

Quadrature formula max accuracy

I'm looking for a maximum accuracy quadrature formula: $$ \int_{-1} ^{1} \sqrt{\frac {1-x}{1+x}} f(x)dx = A_1f(x_1)+A_2f(x_2)+R(f) $$ I don't know exactly if it's Trapezoidal rule which has the ...
2
votes
0answers
132 views

2d quadrature weights for an arbitrary set of nodes

I need to estimate the value of a 2d integral $\int_{y_{min}}^{y_{max}}dy \int_{x_{min}}^{x_{max}} dx \, f(x,y) P(x,y)$ I have the an explicit analytical form for $P(x,y)$. I have samples of the ...
2
votes
0answers
986 views

Area Under Generalized Parabolas and Hyperbolas without Calculus

This is shorter and more specific version of certain questions about a rather simple quadrature method. The answers I got were great but not what I asked. The terms in the title for $y=x^p$ look ...
12
votes
5answers
2k views

Integrating Powers without much Calculus

I'll jump into the question and then back off into qualifications and context Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) ...
6
votes
1answer
662 views

Approximation of an integral of a concave function

I suspect this is a homework question somewhere, but I've not seen it elsewhere and it seems like it should be easy: let $f(x)$ be a concave function from $[0,1]$ to the reals such that $f(0) = f(1) =...
1
vote
1answer
387 views

gaussian quadrature

Gaussian quadrature allows us to integrate polynomials up to order $2 n-1$ using only $n$ function values. $\int_{x_0}^{x_1} ( \sum_{i=0}^{2 n-1} a_i x^i ) dx = f(a_0, \dots , a_{2 n-1}) $ thus, the ...