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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26 votes
4 answers
10k views

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} K_{-\nu-5/2}(\lambda\sqrt{\kappa^2+...
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26 votes
3 answers
4k views

What's the state-of-the-art in highly oscillatory integral computation?

What's the state-of-the-art in the approximation of highly oscillatory integrals in both one dimension and higher dimensions to arbitrary precision?
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 ...
0 votes
2 answers
205 views

Adaptive numerical integration of a univariate vector integrand

Background & Problem formulation I'm trying to write a simple program in C++ that performs adaptive numerical integration of vector valued integrands (in one variable), i.e. $$\int_a^b \bar{f}(...
13 votes
5 answers
5k views

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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13 votes
1 answer
646 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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12 votes
1 answer
355 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 ...
  • 501
11 votes
3 answers
698 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}...
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10 votes
2 answers
2k views

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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10 votes
3 answers
452 views

Integrating Lagrange polynomials with many nodes, round-off

Given a set of points $\{x_j\}_{j=1}^n$ in $[-1, 1]$, I would like to compute $$ \int_{-1}^{1} L_i(x)\,\text{d} x $$ exactly. $L_i$ is the Lagrange polynomial with respect to the points $x_j$ with $...
9 votes
1 answer
682 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 ...
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8 votes
1 answer
4k 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 $(1,1000)$....
8 votes
1 answer
2k 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 compute ...
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7 votes
4 answers
28k views

Plot integral function with scipy and matplotlib

I want to plot a numerical integral function of some function $f$ using scipy and matplotlib. How can I do this? I tried the ...
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7 votes
2 answers
3k 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 \...
6 votes
0 answers
282 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 need ...
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4 votes
1 answer
817 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. ...
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2 votes
2 answers
636 views

computing Newton-Cotes weights

For the closed Newton-Cotes quadrature over $[x_1, x_n]$, the coefficients $H_{n,i}$ for $$ \int_{x_1}^{x_n} f(x)\:\text{d}x = h \sum_{i=1}^n H_{n,i} \; f(x_i) $$ are given explicitly by $$ H_{n,r+1} =...
1 vote
0 answers
13k 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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0 votes
1 answer
97 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 $L^2$ ...
0 votes
2 answers
412 views

Calculating integrals for a function approximated by Chebyshev polynomials

Setup (complete, but all very standard): My problem is how to best calculate the cumulative integral of a function which comes out of Spectral Collocation with a chebyshev basis. Take some function $...
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