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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Computing expectations

I want to compute the following conditional expectation $E_{t}[\phi(A_{t+1}, \eta_{t+1})| A_t]$ where $\log A_{t}=\rho \log A_{t-1} + e_{t}$ and $e_{t}$ is IID $N~(0,\sigma_e)$ and $\eta_{t}$ is ...
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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 ...
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1answer
150 views

Numerical evaluation of an elliptic integral in python

Goal: I need to evaluate numerically an integral of the following form: $$ \int_0^\infty \frac{dx}{(a^2+x)\sqrt{(a^2+x)(b^2+x)(c^2+x)}} $$ where $a,b,c \in \mathbb{R}$ are in the interval ...
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147 views

Why does Matlab's integral outperform integrate.quad in Scipy?

I am experiencing some frustration over the way matlab handles numerical integration vs. Scipy. I observe the following differences in my test code below: Matlab's version runs on average 24 times ...
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1answer
43 views

Numerical Principal Value Integration - Hilbert like

I'ld like to calculate the PV of an integral with the form $$ \tilde{G}_l(\omega) = -\frac{2\omega}{\pi} PV\int_0^\infty \frac{\tilde{G}_d(\omega^\prime)}{\omega^2 - {\omega^\prime}^2}d\omega^\prime$$ ...
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Double integrals and parameters in GSL

I'm trying to calculate a numerical integral in C. I have a function that depends on several variables, two of which I want to integrate over while leaving rest of them as parameters. I was going to ...
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72 views

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 ...
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2answers
107 views

(Fortran) Integrating/summing over complicated 3D domain

I have some function $F(k_x,k_y,k_z)$ that I wish to numerically integrate over a polygon domain - physically, I am integrating over the first Brillouin Zone (BZ) of the FCC lattice (a truncated ...
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136 views

Using scipy.quad to calculate difficult integral

When evaluating the integral below in python using scipy.quad I get the following warning: UserWarning: The maximum number of subdivisions (50) has been achieved. If increasing the limit yields no ...
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2answers
110 views

How to implement Gauss-Laguerre Quadrature in Python?

To get the hang of Gauss-Laguerre integration I have decided to calculate the following integral numerically, which can be compared to the known analytical solution: \begin{align} \int_0^{\infty} ...
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150 views

Is there a Gauss-Laguerre integration routine in Python?

I am reading the book "Numerical Recipes in Fortran 77: The Art of Scientific Computing" (Second Edition) and I came across some methods for numerical integration of 1D functions. More specifically ...
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95 views

Tanh-sinh quadrature numerical integration method converging to wrong value

I'm trying to write a Python program to use Tanh-sinh quadrature to compute the value of \begin{equation} \int_{-1}^1 \frac{dx}{\sqrt{1-x^2}} \end{equation} but although the program converges to a ...
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1answer
47 views

How do error estimates scale for multidimensional cubature?

Suppose I have a quadrature method with a theoretical error estimate that scales with the number of points as $e(N)$: $$\int_a^b f(x)\mathrm{d}x = \sum_{i=1}^N w_i f(x_i) + ...
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3answers
114 views

Surface integration over a portion of an ellipsoid

I would like to perform a surface integration over a portion $D$ of an ellipsoid. A plane arbitrarily intersects the ellipsoid forming two sections, of which one is $D$. I do not know how I can ...
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1answer
263 views

How do I integrate this function in python?

Essentially this is the problem: $\hat{F}(\omega) = \int_0^{\infty} f(s)e^{-i\omega s}ds$ The function $f$ is in general complex valued. I know this looks like the fourier transform but I don't want ...
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1answer
155 views

Solving the quadratic in the Fast Marching Method

The Fast Marching Method is a way of solving the Eikonal Equation on a discrete grid, essentially just computing a wavefront speading out from initial points, IE: The idea is that we want to ...
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3answers
105 views

Fast way to compute integral of type $\int dx f(x) \cos(n \pi x)$ in SciPy

I have an integral of the form $$ I(n) = \int_0^1 dx f(x) \cos(n \pi x) , $$ where $n$ is an integer. In other words, I calculate the cosine Fourier coefficients of function $f$, which is real and ...
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97 views

Numerical spherical integration

In a high-dimensional setting, say $d \gg 5$, what is a recommended way of evaluating a spherical integral of a smooth (non-symmetric) function $f(\mathbf{x})$? $ \int_\mathcal{S_r} f(\mathbf{x}) ...
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157 views

Numerical Integration with Convergence Factor with SciPy: Problem with Improper Integral

I would like to perform the numerical integration of an integral of the form $$ \int_{-\infty}^\infty e^{i \omega 0+} G(i \omega, \mathbf{v}) d \omega ,$$ or, using the symmetry $G(i\omega)^* = G(-i ...
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Radial integration of expensive function with Bessel weights

I need to calculate the integral $$I = \int_0^R f(r)J_n\left(\frac{z_{nm}r}{R}\right)rdr$$ where $J_n$ is the $n^{\mathrm{th}}$ order Bessel functions of the first kind, $z_{nm}$ is its ...
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Numerical Quadrature of Oscillating Integral With Non-oscillating part

As you will know there are different numerical integrals (I believe Levin's method is the most popular one) for the numerical quadrature of oscillating integrands which may roughly speaking be written ...
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Numerical integral with a weakly singular kernel with a satisfactory precision

I am working on numerical method for time fractional PDE. One problem is that I must compute numerical integral with the following form: \begin{equation} \int_0^{t_m} (t_m-s)^{-\beta}f(s)ds ...
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Methods for integrating black box functions on a non-uniform grid

If i have some function expressed as points on a non-uniform grid (I'm specifically interested in logarithmic grids, but general results are also interesting), and I want to integrate it, I believe ...
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128 views

Trapezoidal rule (linear Newton-Cotes integration) convergence for periodic functions on general non-uniform grids

For periodic functions on a regularly spaced grid, the trapezoidal rule is supposed to converge incredibly quickly ( the error estimate decreasing as $O(e^{-\eta/h})$ for step size $h$ ). However, I ...
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Integrating highly oscillatory functions

I have a logarithmic grid, upon which i have two functions that are similar to this one (this is only the last 100 points): These are essentially very similar to a Sin function at this point. I ...
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116 views

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 ...
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297 views

adaptive Gauss-Kronrod quadrature with vector-valued integrand

So I'm trying to implement a Gauss-Kronrod adaptive quadrature. That is, I want to calculate $$\int_a^b f(x) dx = \sum_i f(x_i) w_i$$ where f(x) is evaluated at multiple points at once for ...
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191 views

How to integrate numerically over a radial domain

I want to integrate a function over a radial domain $D=\{r<r(\theta)\}$. The change to polar coordinates yields: $$ \int_D f = \int_0^{2\pi} \int_0^{r(\theta)}f(r,\theta)rdr d\theta $$ so I tried ...
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581 views

Integration of matrix-valued function using MATLAB

I need to compute the integral $\int_{\mathbb{R}^3}f(x)\, dx$, where $f$ is a matrix-valued function $f:\mathbb{R}^3\to\mathbb{R}^{3\times 3}$. How do I do that using MATLAB? The function $f$ is ...
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315 views

How can I get Octave to correctly compute a double integral of a piecewise function?

Computational Science People: The title is the question. I am trying to numerically compute a certain integral over a square in Octave (an open-source Matlab clone). I'm getting the wrong answer. ...
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FEniCS: custom quadrature rule

For the numerical integration of reaction terms in my PDE on a 2D triangular mesh, I would like to use the scheme $$ \int_T \Phi(x) \approx \frac{|T|}{3} (\Phi(x_0) + \Phi(x_1) + \Phi(x_2)) $$ where ...
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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 ...
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1answer
170 views

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 ...
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In C++, how do you calculate the analytical value of $\int_a^b \left|\sin x \right|\,dx$?

How would I find the definite integral (between any 2 limits, say a and b) of the absolute value of sin(x)? I can calculate for the interval 0 to Pi, and from 0 to 2*Pi, but what if the user enters a ...
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391 views

Numeric Quadrature with Derivatives

Most numerical methods for quadrature treat the integrand as a black-box function. What if we have more information? In particular, what benefit, if any, can we derive from knowing the first few ...
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125 views

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 ...
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1answer
868 views

Finite Element Method: 2-D Poisson's Equation in Matlab, Gaussian quadrature

I'm having trouble understanding how to code 2-D Poisson's Equation with Dirichlet boundary conditions. What I have thus far is Constructed square mesh with triangular elements Assembled stiffness ...
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226 views

Does transforming $J_0(x)\to\int\cos(x\sin\theta)$ help with numerical integration?

I've heard anecdotally that when one is trying to numerically do an integral of the form $$\int_0^\infty f(x) J_0(x)\,\mathrm{d}x$$ with $f(x)$ smooth and well-behaved (e.g. not itself highly ...
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Method for numerical integration of difficult oscillatory integral

I need to numerically evaluate the integral below: $$\int_0^\infty \mathrm{sinc}'(xr) r \sqrt{E(r)} dr$$ where $E(r) = r^4 (\lambda\sqrt{\kappa^2+r^2})^{-\nu-5/2} ...
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397 views

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 ...
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1answer
225 views

How to integrate polynomial expression over 3D 4-node element?

I want to integrate a polynomial expression over a 4-node element in 3D. Several books on FEA cover the case where integrating is performed over an arbitrary flat 4-noned element. The usual procedure ...
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3answers
715 views

Numeric integration of multi-dimensional integral with known boundaries

I have a (2-dimensional) improper integral $$I=\int_A \frac{W(x,y)}{F(x,y)}\,\mbox{d}x\mbox{d}y$$ where the domain of integration $A$ is smaller than $x=[-1,1]$, $y=[-1,1]$ but further restricted by ...
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485 views

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 ...
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How can I approximate an improper integral?

I have a function $f(x,y,z)$ such that $\int_{R^3} f(x,y,z)dV$ is finite, and I want to approximate this integral. I'm familiar with quadrature rules and monte carlo approximations of integrals, ...
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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 ...
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272 views

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 ...
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1answer
611 views

Error calculation in trapezoidal rule

If we use the composite trapezoidal rule, then what is the least number of divisions $N$ for which the error of the integral $\int^1_0{e^{-x}}dx$ doesn't exceed $\frac{1}{12}\times10^{-2}$. My guess ...
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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 ...
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181 views

Quadrature rules, methodologies, and references

There is at least one quite comprehensive encyclopaedia of quadrature rules that doesn't seem to have been updated in quite a while and has restricted access. This source refers to several classical ...
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348 views

What numerical quadrature to choose to integrate a function with singularities?

For example, I would like to numerically compute the $L^2$-norm of $\displaystyle u = \frac{1}{(x^2+y^2+z^2)^{1/3}}$ in some domain that includes zero, I tried Gauss quadrature and it fails, it is ...