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Questions tagged [quadrature]

Also called numerical integration, quadrature refers to the approximation of an integral made by evaluating the integrand at a finite number of points.

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Are there standard methods for joining a numerical function approximation to its asymptotic behavior?

I'm working on an algorithm for computing a function that is a generalization of the Voigt profile. The Voigt profile is the convolution of a Cauchy (aka Lorentzian) probability density function (PDF) ...
Tom Loredo's user avatar
2 votes
0 answers
62 views

Understanding proof of the error bound for Simpson's quadrature rule

I have found the following proof of the error bound for Simpson's quadrature rule: Using Newton's interpolation method, we derive a cubic polynomial $p_3(x)$ that interpolates $f(x)$ at the points $a, ...
codeing_monkey's user avatar
2 votes
0 answers
98 views

Calculating Debye functions to high accuracy (hundreds of bits), is it possible to be faster than generic quadrature?

The Debye functions are defined like so: ${D_n\left(x\right)} = {\frac{n}{x^n} \cdot {\int_0^x{\frac{t^n}{e^t - 1}dt}}}$. I'm trying to evaluate the functions for $n$ from one to four and for $\left\...
user2373145's user avatar
1 vote
0 answers
59 views

3D Quadrature schemes with points on boundary

In one dimension there are two types of quadrature schemes. asymmetric rules like Newton-Cotes like formulas (Trapezodi, Simpson), and Clenshaw-Curtis place sampling points on boundary of the ...
Prokop Hapala's user avatar
3 votes
1 answer
167 views

Optimal quadrature rule for heavy tail measure

I'm looking for a well-thought quadrature rule for this measure $d\mu(t)=\frac{dt}{t^s}$ for $s\in(0,1)$, the underlying motivation is to compute this integral $$ \lambda^{s-1}=\frac{1}{\Gamma(1-s)}\...
Aner's user avatar
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2 votes
0 answers
78 views

Quadrature rules with the weight function $w(x) = |x|^\gamma$

I am interested in integrals of the form $$ \int_{[0,1]^{d}} |x|^{\gamma}f(x)dx. $$ $\gamma>0$ and $f$ has some singular behavior at $\vec{0}$. The weight function $|x|^\gamma$ is commonly used in ...
Justin Dong's user avatar
1 vote
0 answers
96 views

Integration problem

I want to numerically solve integrals of the form, $$ I = \int_a^b x^k f(x) dx $$ where $k$ is a given integer, and $f(x)$ is a cubic polynomial, expressed as, $$ f(x) = c_0 + c_1 (x - a) + \frac{c_2}{...
vibe's user avatar
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0 answers
44 views

Quadrature rules for products of 2D regions

I am interested in computing integrals of the form $\iint_{P\times P} Q(x_1,x_2,y_2,y_2) dxdy$ where $P$ is a polygon and $Q$ is a polynomial. The coordinates $(x_1,x_2)$ are in the plane of $P$. Of ...
Beni Bogosel's user avatar
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5 votes
2 answers
211 views

Which way is the right way to compute the integrals in finite element methods?

Finite element methods involve integrals of functions that are not polynomials, and these integrals must be computed numerically. For example, suppose that $f$ is the right-hand side of a Poisson ...
shuhalo's user avatar
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2 votes
1 answer
196 views

Quadrature of rational functions

I have a class of integrals I need to solve numerically which have the form: $$ I_k = \int_a^b \frac{p_k(x)}{x^k} dx, \quad k = 0, 1, \dots, K $$ where $p_k(x)$ is a cubic polynomial on the interval $[...
vibe's user avatar
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1 vote
1 answer
109 views

numerical integration of integrals in the p-adaptive version of the finite element method

In the p-adaptive version of the finite element method, elements are allowed to have shape functions with arbitrary different polynomial orders. Therefore regarding a 2D problem with quadrilateral ...
Masa's user avatar
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4 votes
0 answers
142 views

Quadrature rules for non-linear finite element problems

For solving linear problems stemming from PDEs with the FEM, such as the Poisson equation or the wave equation, it is customary to use the "simplest" numerical quadrature that exactly ...
Andreas Longva's user avatar
6 votes
3 answers
270 views

Numerical integration giving incorrect sign

For my research, I need to integrate the following function: $$ W(z)=\int_0^{\infty}dx\ w(x,z)\\ =\int_0^{\infty}dx\frac{e^x}{(e^x+1)^2}\log{\left(\frac{e^{z^2/4x+x}+1}{e^{z^2/4x+x}-e^x}\right)}\\ =\...
surrutiaquir's user avatar
5 votes
2 answers
195 views

Optimizing a quadratic form integral over unit sphere

I have an optimization problem, which is to maximize the following integral over the unit sphere: $$ \max_B \int d\Omega \mathbf{f}^{\dagger}(\theta,\phi) (B^{\dagger} + B) \mathbf{f}(\theta,\phi) $$ ...
user3516849's user avatar
2 votes
1 answer
177 views

Variational loss of hp-Variational Physics Informed Neural Networks for 2D-Poisson Equation in Tensorflow

I am trying to reproduce the results from the hp-VPINN paper (https://arxiv.org/pdf/2003.05385.pdf) on tensorflow (v1) for Poisson's equation, particularly the two-dimensional Poisson equation. In one ...
M.V.'s user avatar
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1 answer
109 views

How is the integral of a projection over an element $T$ computed in practice? (deal.II related)

I'm studying an error estimator for the equation $\nabla\cdot(\beta u) + cu = f$ and it contains the following term $$||f - cU_h - \Pi(f-c U_h) ||_T$$ where : $\Pi$ is the local orthogonal $L^2$ ...
FEGirl's user avatar
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1 vote
1 answer
403 views

Compute 2D numerical double integration with Boost C++ with parameters

I am trying to compute the double Richardson and Wolf integrals for the focusing of a lens with Boost in C++ (using the Gauss Kronrod method). As a starting point, I used the example presented in this ...
Bertrand Simon's user avatar
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43 views

Calculating magnetic flux density

I have a geometry where current density distribution is constant. I can calculate the $z$ component of magnetic flux density according to Biot-Savart law as following: $$B_z(x,y,z) = \int\limits_x\int\...
strahd's user avatar
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-1 votes
1 answer
1k views

Integrate function with cumulative distibution function inside

I'm trying to integrate a function which is defined as func in my code below, a cumulative distribution function is inside: ...
maliniaki's user avatar
6 votes
1 answer
2k views

Gauss-Lobatto quadrature and nodal points for FEM

By using the Legendre-Gauss-Lobatto (LGL) quadrature formula (QF) and LGL nodal points one achives a diagonal mass-matrix for finite element problems. (More specifically, the spectral element method.) ...
Dagon's user avatar
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-1 votes
1 answer
123 views

Integral over a surface, given experimental data

I have a mesh of a 3D surface composed by triangles, and I have the value of a function $u(x,y,z)$ in every vertex of the mesh (every vertex of each triangle). I need to calculate the following ...
yemino's user avatar
  • 515
3 votes
1 answer
264 views

Find quadrature points and weights

I'm struggling with the following problem: What is the maximum degree of exactness that we can obtain with the following quadrature >formula $$\int_0^1 f(x)\frac{1}{\sqrt{x}}dx \approx w_0 f(x_0) +...
lukk's user avatar
  • 31
3 votes
1 answer
658 views

How to use numerical integration to calculate the surface area of a superellipsoid?

I am working in an application in which I need to calculate the surface area of a superellipsoid. I have read that there is no closed form solution (see here), so I am trying to compute it using ...
llorente's user avatar
2 votes
3 answers
192 views

Numerical solution of high-dimensional integral involving positive-part function

Consider a potentially high-dimensional (say, $N$ up to 20) integral of the form $$ \int_0^\infty \rho_1(x_1)\rho_2(x_2) \cdots \rho_N(x_N) \bigg(x_1+x_2+\cdots+x_N -K\bigg)^+ \, dx_1 \cdots dx_N. $$ ...
davidhigh's user avatar
  • 3,147
0 votes
0 answers
69 views

Adaptive quadrature methods for Fourier Optics

In Fourier Optics one often needs to compute approximations to bivariate integrals like $$ \int_{-\frac{l}{2}}^{\frac{l}{2}}\int_{-\frac{l}{2}}^{\frac{l}{2}} {\rm e}^{i\phi(\xi,\eta)}\mathrm{exp}\left[...
Arrigo's user avatar
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1 vote
0 answers
164 views

Is Romberg integration method implemented as weighted function values numerically correct?

I have to integrate expression f(x) * g(x) for many different functions f but just one g. I ...
abukaj's user avatar
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2 votes
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52 views

To use the confluent hypergeometric function or not to?

I am numerically computing the following integral as a function of positive $k$. $$I(k) := \int_0^\infty x^b(k+x)^{a-1} e^{-x} dx \tag1$$ It is shown in math.stackexchange.com that this can be ...
Hans's user avatar
  • 121
4 votes
1 answer
1k views

Performing 2d numerical integration with Boost Cpp

I've been learning to use the numerical quadrature of the Boost library for Cpp. In the documentation, I've found an example for 1D Gauss-Kronrod Quadrature using Boost. ...
Galilean's user avatar
  • 151
2 votes
1 answer
838 views

How to implement Simpson's rule for double integral (without numeric limits of first integral)

I want to use Simpson's rule to evaluate the following double integral: $$\int_{a}^{b}\left|\int_{0}^{z}x\cdot \mathrm{erf}(x-10)\cdot J_{0}(x) \mathrm{dx}\right|^{2}\exp(-0.5*(z-40)^2)\mathrm{dz} $$ ...
Shankar_Dutt's user avatar
1 vote
1 answer
121 views

Evaluating an indefinite integral that has no closed form

I need to evaluate the following indefinite integral: $$I=\int\frac{x^5+2ax^3+a^2x-4a}{x^7+ax^5+2ax^4}dx=\int\frac{x^5+2ax^3+a^2x-4a}{x^4(x^3+ax+2a)}dx$$ The solution that I obtained while ...
Richard's user avatar
  • 131
1 vote
1 answer
92 views

Algorithm for evaluation of spin-weighted spherical harmonics

Is there an algorithm to evaluate spin-weighted spherical harmonics (swSH) at arbitrary points on the sphere? In particular I am looking for, e.g. a recursion relation to evaluate the "spin weighted ...
physics_researcher's user avatar
7 votes
1 answer
243 views

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 ...
Michael Anderson's user avatar
4 votes
2 answers
8k views

Numerical evaluation of a Gaussian Integral in Python?

Goal I'm trying to write code to compute the normalized Gaussian in the following, $$ \begin{equation} \int_{-\infty}^{\infty} \frac{1}{ \sigma \sqrt{2 \pi}} \exp\bigg( - \frac{(x - \mu)^{2}}{2 \...
Zophikel's user avatar
  • 143
11 votes
3 answers
940 views

Numerical evaluation of highly oscillatory integral

In this advanced course on applications of complex function theory at one point in an exercise the highly oscillatory integral $$I(\lambda)=\int_{-\infty}^{\infty} \cos (\lambda \cos x) \frac{\sin x}...
doetoe's user avatar
  • 593
-2 votes
1 answer
113 views

How to : numerical integration by quadrature in C language / remove NaN

What I wanna solve it the problem following ( by quadrature method ) I want to get two arrays of data ( z & tau ) from z[0], tau[0] to z[2249], tau[2249]. Since the integrand diverges at z=0.9, ...
Quantum John Do's user avatar
1 vote
0 answers
48 views

How to integrate the contents of a vector using an adaptive quadrature routine [duplicate]

I have a function which requires the return type to be a container. The problem is that I need to integrate the contents of the container as efficiently as possible and was hoping to use adaptive ...
AlexD's user avatar
  • 131
2 votes
0 answers
62 views

Does adaptive Gauss-Kronrod reuse function evaluations?

I'm curious to know how QUADPACK's QAG routine works. My understanding is that it begins by calculating on each subinterval the numerical quadrature with a Gaussian-Legendre rule and a nested Kronrod ...
Ian's user avatar
  • 161
1 vote
1 answer
29 views

Domain transformation squashing interior quadrature nodes into boundary

In many quadrature problems, we are interested in computing $\int_a^b f(x) \, \mathrm{d}x$ via a quadrature sum. However, most software packages precompute the quadrature nodes and weights for use ...
user14717's user avatar
  • 2,155
7 votes
0 answers
96 views

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 ...
Endulum's user avatar
  • 735
0 votes
1 answer
208 views

How to cope with the following singularity

I have the following integral: $\int_{1}^{Xd} \dfrac{(X^{z_i}-1)}{[X^2 \sum_{l=1}^{N}c_l(X^{z_l}-1)]^{1/2}}dX = \int_{1}^{Xd} h(X) dX$ where: Xd is a real that can be either negative, positive or ...
Daniel's user avatar
  • 99
6 votes
1 answer
432 views

How do I integrate a function defined over an arbitrary area?

Let's say, I have a compact area $S$ (for example a circle, a square or some arbitrary polygon) and a function $f: S \rightarrow \mathbb{R}$. I want to numerically calculate the Integral $$ \int_S f(\...
HerpDerpington's user avatar
1 vote
2 answers
344 views

How many quadrature points should I use?

I am trying to compute the following integration $$ \int_0^\infty e^{-y}y^{a/2}L_c^b(y)L_e^d(y)dy $$ using the generalized Gauss-Laguerre quadrature routine in the GNU Scientific Library. Here the $L$'...
e-eight's user avatar
  • 163
1 vote
0 answers
30 views

Algorithm for integrating a 6D function in a Morse-Smale 3D cell

Lets say that one has a scalar field defined in 3D space for whose gradient he wants to find the Morse-Smale Complex for later performing an integration of several hexa-dimensional functions over ...
Zythos's user avatar
  • 181
5 votes
1 answer
345 views

What is the best numerical method for a six dimensional spherical integral?

I am trying to do integrals of the type $$ \int d^3\vec{p} \int d^3\vec{p}' e^{-p^2} e^{-{p'}^2}f(\vec{p}, \vec{p}') $$ where $\vec{p}$ and $\vec{p}'$ are three dimensional vectors represented using ...
e-eight's user avatar
  • 163
7 votes
3 answers
267 views

Evaluating an integral numerically at many points

Given a real function $f$, how can one efficiently evaluate $\int_0^{a_i}f(x)dx$ for millions of different $a_i$? Applying a standard quadrature method (such as Simpson's rule or Gaussian quadrature) ...
Museful's user avatar
  • 255
12 votes
2 answers
428 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, ...
Wrzlprmft's user avatar
  • 2,022
3 votes
0 answers
334 views

Convergence of Gauss quadrature for a discontinuous function

Is there a known error estimate for Gaussian quadrature when applied to a discontinuous function? For simple one-dimensional experiments, the error appears to be bounded by $C h$, where $C$ is some ...
Jesse Chan's user avatar
  • 3,142
1 vote
0 answers
53 views

Stability of different quadrature rules in 1st-kind Volterra integral equation

I am dealing with a integral equation $$ f'(t) = -\int_0^t K(s) f(t-s)\quad t\in [0,t_\max] \tag{1}$$ in which $f(t)$ and $f'(t)$ are known, well-behaved functions of $t$ and $K(t)$ is the unknown. In ...
Endulum's user avatar
  • 735
5 votes
1 answer
172 views

Kernel-based differentiation

Consider a $\mathcal{C}^1$ function $V:\Omega\rightarrow\mathbb{R}$ where $\Omega\subset\mathbb{R}^n$. If a random vector $X$ has a parametric density $p_\theta(\textbf{x})$ that's smooth in its ...
VF1's user avatar
  • 211
5 votes
2 answers
266 views

Building Gaussian-type quadrature schemes with Zernike polynomials

The abscissas for Gauss quadrature are given by the zeros of the Legendre polynomials. The Legendre polynomials form an orthogonal set over $[-1, 1]$, and it is shown in (for instance) Kress that the ...
user14717's user avatar
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