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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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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 $(1,1000)$....
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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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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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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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(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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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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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} s^...
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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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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 $$\int_{-1}^1 \frac{dx}{\sqrt{1-x^2}}$$ but although the program converges to a ...
Suppose I have a quadrature method with a theoretical error estimate that scales with the number of points as $e(N)$: $$\int_a^b f(x)\mathrm{d}x = \sum_{i=1}^N w_i f(x_i) + \mathcal{O}\bigl(e(N)\bigr)... 3answers 307 views Surface integration over a portion of an ellipsoid I would like to perform a surface integration over a portion D of an ellipsoid. A plane arbitrarily intersects the ellipsoid forming two sections, of which one is D. I do not know how I can ... 1answer 378 views How do I integrate this function in python? Essentially this is the problem: \hat{F}(\omega) = \int_0^{\infty} f(s)e^{-i\omega s}ds The function f is in general complex valued. I know this looks like the fourier transform but I don't want ... 1answer 534 views Solving the quadratic in the Fast Marching Method The Fast Marching Method is a way of solving the Eikonal Equation on a discrete grid, essentially just computing a wavefront speading out from initial points, IE: The idea is that we want to ... 3answers 148 views Fast way to compute integral of type \int dx f(x) \cos(n \pi x) in SciPy I have an integral of the form$$ I(n) = \int_0^1 dx f(x) \cos(n \pi x) , $$where n is an integer. In other words, I calculate the cosine Fourier coefficients of function f, which is real and ... 1answer 118 views Numerical spherical integration In a high-dimensional setting, say d \gg 5, what is a recommended way of evaluating a spherical integral of a smooth (non-symmetric) function f(\mathbf{x})?  \int_\mathcal{S_r} f(\mathbf{x}) \... 2answers 645 views Numerical Integration with Convergence Factor with SciPy: Problem with Improper Integral I would like to perform the numerical integration of an integral of the form$$ \int_{-\infty}^\infty e^{i \omega 0+} G(i \omega, \mathbf{v}) d \omega ,$$or, using the symmetry G(i\omega)^* = G(-i \... 0answers 45 views Radial integration of expensive function with Bessel weights I need to calculate the integral$$I = \int_0^R f(r)J_n\left(\frac{z_{nm}r}{R}\right)rdr$$where J_n is the n^{\mathrm{th}} order Bessel functions of the first kind, z_{nm} is its m^{\mathrm{... 0answers 72 views Numerical Quadrature of Oscillating Integral With Non-oscillating part As you will know there are different numerical integrals (I believe Levin's method is the most popular one) for the numerical quadrature of oscillating integrands which may roughly speaking be written ... 0answers 157 views Numerical integral with a weakly singular kernel with a satisfactory precision I am working on numerical method for time fractional PDE. One problem is that I must compute numerical integral with the following form: \int_0^{t_m} (t_m-s)^{-\beta}f(s)ds \end{... 0answers 69 views Methods for integrating black box functions on a non-uniform grid If i have some function expressed as points on a non-uniform grid (I'm specifically interested in logarithmic grids, but general results are also interesting), and I want to integrate it, I believe ... 1answer 259 views Trapezoidal rule (linear Newton-Cotes integration) convergence for periodic functions on general non-uniform grids For periodic functions on a regularly spaced grid, the trapezoidal rule is supposed to converge incredibly quickly ( the error estimate decreasing as O(e^{-\eta/h}) for step size h ). However, I ... 0answers 136 views Integrating highly oscillatory functions I have a logarithmic grid, upon which i have two functions that are similar to this one (this is only the last 100 points): These are essentially very similar to a Sin function at this point. I ... 1answer 127 views Reliable quadrature software I would like to numerically integrate a smooth function (to be precise, I am computing differential entropies of Gaussian mixtures of the form \int f(x) \log(1/f(x)) dx). Is there any software ... 2answers 673 views adaptive Gauss-Kronrod quadrature with vector-valued integrand So I'm trying to implement a Gauss-Kronrod adaptive quadrature. That is, I want to calculate$$\int_a^b f(x) dx = \sum_i f(x_i) w_i$$where f(x) is evaluated at multiple points at once for ... 1answer 416 views How to integrate numerically over a radial domain I want to integrate a function over a radial domain D=\{r<r(\theta)\}. The change to polar coordinates yields:$$ \int_D f = \int_0^{2\pi} \int_0^{r(\theta)}f(r,\theta)rdr d\theta $$so I tried ... 2answers 1k views Integration of matrix-valued function using MATLAB I need to compute the integral \int_{\mathbb{R}^3}f(x)\, dx, where f is a matrix-valued function f:\mathbb{R}^3\to\mathbb{R}^{3\times 3}. How do I do that using MATLAB? The function f is ... 1answer 688 views How can I get Octave to correctly compute a double integral of a piecewise function? Computational Science People: The title is the question. I am trying to numerically compute a certain integral over a square in Octave (an open-source Matlab clone). I'm getting the wrong answer. ... 2answers 507 views FEniCS: custom quadrature rule For the numerical integration of reaction terms in my PDE on a 2D triangular mesh, I would like to use the scheme$$ \int_T \Phi(x) \approx \frac{|T|}{3} (\Phi(x_0) + \Phi(x_1) + \Phi(x_2))  where ...
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 \...