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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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 ...
5 votes
2 answers
194 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 ...
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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 $[...
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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 ...
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4 votes
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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 ...
6 votes
3 answers
252 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)}\\ =\...
5 votes
2 answers
129 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: $$ \max_B \int d\Omega \mathbf{f}^{\dagger}(\theta,\phi) (B^{\dagger} + B) \mathbf{f}(\theta,\phi) $$ ...
2 votes
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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 ...
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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 $L^2$ ...
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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 ...
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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\...
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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: ...
5 votes
1 answer
931 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.) ...
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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 ...
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1 answer
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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) +...
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439 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 ...
2 votes
3 answers
182 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 $$ \int_0^\infty \rho_1(x_1)\rho_2(x_2) \cdots \rho_N(x_N) \bigg(x_1+x_2+\cdots+x_N -K\bigg)^+ \, dx_1 \cdots dx_N. $$ ...
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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[...
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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 ...
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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 ...
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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. ...
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2 votes
1 answer
556 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: $$\int_{a}^{b}\left|\int_{0}^{z}x\cdot \mathrm{erf}(x-10)\cdot J_{0}(x) \mathrm{dx}\right|^{2}\exp(-0.5*(z-40)^2)\mathrm{dz} $$ ...
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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 ...
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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 ...
7 votes
1 answer
226 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 ...
4 votes
2 answers
6k 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 \...
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11 votes
3 answers
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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}...
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1 answer
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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, ...
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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 ...
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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 ...
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1 vote
1 answer
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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 ...
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7 votes
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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 ...
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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 ...
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6 votes
1 answer
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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(\...
1 vote
2 answers
321 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 $L$'...
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1 vote
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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 ...
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5 votes
1 answer
249 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 ...
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7 votes
3 answers
212 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) ...
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12 votes
2 answers
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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, ...
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3 votes
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253 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 ...
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1 vote
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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 ...
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5 votes
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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 ...
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5 votes
2 answers
220 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 ...
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Modulus (absolute) of a function, its quadrature, and relevance of zeros

Modulus of a (discretized) function, $|f_h(x)|$, where $h$ refers to the mesh spacing, would, in general, have zeros, and those zeros would not necessarily lie exactly at the mesh points. A naive <...
0 votes
1 answer
129 views

Reference for Dunavant Quadrature Implementations

I am using Dunavant quadrature in my software, specifically this file by John Burkardt. Recently, I wanted to convert the code into a constexpr code in C++. But ...
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5 votes
1 answer
1k views

2D numerical integration with infinite limit (C++)

In order to integrate a two dimensional function of the form $$\int_{1}^\infty \int_{-\sqrt{x^2-1}}^{\sqrt{x^2-1}} e^{-x} \rm{d}y \rm{d}x,$$ I have been attempting to use the following code (written ...
3 votes
1 answer
2k views

Line integral along the edge of an isoparametrically mapped triangle

I need to integrate the following function on the line segment from $P_{1} = \begin{bmatrix} -2\\-1 \end{bmatrix}$ to $P_{2} = \begin{bmatrix} 1\\2 \end{bmatrix}$: $$\int_{P_{1}}^{P_{2}} 4x + y \ ds$$...
1 vote
1 answer
2k views

Computation of stiffness matrix with variable coefficient

I am implementing a finite element solver (in 2D) to solve the generic differential equation : $$-\nabla(a(x) \nabla u) = f$$ Brief explanation By integrating and multipling by a test function, the ...
1 vote
0 answers
436 views

Numerical integral of oscillating function with known zeros

I have a function that I need to numerically integrate from $0$ to $+\infty$, given by: $$I = \int_0^{+\infty} \mathrm{d}x\,x\,T^2(x)f(x)$$ where $T^2$ is an interpolated function that goes to $1$ ...
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190 views

Spherical volume integral from pre-calculated points - which algorithm is best?

I need a fast and accurate method to calculate 3d spherical volume integrals. I have pre-calculated data of high precision that just needs a few trivial manipulations before each integration step - ...
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