Also called numerical integration, quadrature refers to the approximation of an integral made by evaluating the integrand at a finite number of points.
Questions tagged [quadrature]
143 questions
13
votes
1 answer
Method selection for numeric quadrature
Several families of methods exist for numeric quadrature. If I have a specific class of integrands how do I select the ideal method?
What are the relevant questions to ask both about the integrand (e.g. is it smooth? does it have singularities?)…
MRocklin
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12
votes
2 answers
Numerical Integration - handling NaNs (C / Fortran)
I am dealing with a tricky integral that exhibits NaNs at certain values near zero and at the moment I am dealing with them quite crudely using an ISNAN statement which sets the integrand to zero when this occurs. I have tried this with the NMS…
Josh
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11
votes
1 answer
Integrating a harmonic function over a tetrahedron
Say I have a function $f : \mathbf{R}^3 \to \mathbf{R}$ that I wish to integrate over a tetrahedron $T \subset \mathbf{R}^3$. If $f$ was arbitrary, Gauss quadrature would be a good solution, but I happen to know that $f$ is harmonic. How much can…
Geoffrey Irving
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10
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Higher-order numerical integration on a triangle/tetrahedron/simplex
Let $T$ be a triangle and let $f$ be a smooth function on $T$.
We can use mid-point quadrature $\int f dx \approx |T|\cdot f(x_M)$, where $x_M$ is the middle-point of $T$.
Can you provide me with (a reference for) higher-order formulas on a simplex?
shuhalo
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10
votes
3 answers
Numerical integration of compactly supported function on a triangle
as the title suggests I'm trying to compute the integral of a compactly supported function (Wendland's quintic polynomial) on a triangle. Notice, that the center of the function is somewhere in 3-D space. I integrate this function on an arbitrary,…
Azrael3000
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9
votes
2 answers
Evaluating oscillatory integrals with many independent periods and no closed forms
Most methods for oscillatory integrals I know about deal with integrals of the form
$$ \int f(x)e^{i\omega x}\,dx $$
where $\omega$ is large.
If I have an integral of the form
$$ \int f(x)g_1(x)\cdots g_n(x)\,dx, $$
where $g_k$ are oscillatory…
Kirill
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9
votes
1 answer
Suggestions for numerical integral over Pólya Distribution
This problem arises from a Bayesian statistical modeling project. In order to compute with my model, I need to perform an integration in which part of the integrand is the "Pólya" or "Dirichlet-Multinomial" Distribution,
$$p(n\mid \alpha) =…
yep
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7
votes
1 answer
Numerical calculation of Integral of Si(x)/x
I'm interested in evaluating
\begin{equation}
\int_0^x \frac{Si(t)}{t}\;dt
\end{equation}
Where
\begin{equation}
Si(x) = \int_0^x \frac{\sin t}{t}\;dt
\end{equation}
I've found a nice method for evaluating $Si(x)$ using Pade and Chebyshev-Pade…
Michael Anderson
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Quadrature methods for peaky integrands
Consider
$$
I = \int_{-L}^L f(x)dx,
$$
where $f(x)$ is real-valued and analytic on $[-L,L]$, but it has a pole in the complex plane whose real part lies in $[-L,L]$. Call it $z_0$, and assume it is a simple pole. If the imaginary part of $z_0$ is…
Endulum
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6
votes
1 answer
Numerical computation of two-sided (bilateral) Laplace transform
I need to compute the two-sided (bilateral) Laplace transform of a numerically given function $F$,
$$
I(t) = \int_{-\infty}^{+\infty} {dx} \, e^{-x} \, F(x + t) ~,
$$
where $F(x)$ has some sharp features, e.g., at $ x = \{ 0, x_p, \cdots \} $, has…
AlQuemist
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5
votes
1 answer
How to integrate numerically over a radial domain
I want to integrate a function over a radial domain $D=\{r
Beni Bogosel
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votes
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Finding quadrature weights for a given set of points? How to select points such that all weights are positive?
Currently, I fit a Finite Element solution of a PDE on a spectral basis. The matrices ($R^{25000\times 2000}$) of the corresponding system of linear equations are highly ill-conditioned ($\kappa \approx 10^{15}$). However, I am able to find a…
Bort
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5
votes
2 answers
Methods to solve a double integral
I want to solve the following expression (used to obtain an analytic solution to a current distribution inside a workpiece):
$$a_{mn} = -\frac{\frac{4}{ab} \int_0^a \int_0^b f(x',y')\sin(px')\sin(qy')\mathrm{d}x'\mathrm{d}y'}{t\sinh(tc)}$$
Here.…
akid
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votes
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Comparison between Voronoi and Delaunay 2D quadrature methods
This question is a search for further answers from a question on maths.stackexchange.com.
I've inherited some numerical quadrature code that is designed to integrate sparse 2D data. The quadrature is not being done in any of what I would consider…
Michael Anderson
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4
votes
1 answer
Reliable quadrature software
I would like to numerically integrate a smooth function (to be precise, I am computing differential entropies of Gaussian mixtures of the form $\int f(x) \log(1/f(x)) dx$).
Is there any software package or library that gives me the exact answer up…
numerical-integrator
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