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.

21 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
8
votes
1answer
222 views

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 "Dirichlet-...
6
votes
0answers
69 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 ...
6
votes
0answers
260 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 ...
5
votes
0answers
186 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 ...
5
votes
0answers
187 views

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 \...
4
votes
0answers
141 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 ...
3
votes
0answers
173 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 ...
3
votes
0answers
382 views

Calculate Integral Using Gauss Jacobi Quadrature or otherwise

I need to integrate the following integral: \begin{align} I = \int^z\frac{1-\zeta^2}{(1+\zeta^2)(\zeta-\zeta_l)(1-\zeta_l\zeta)}\prod_{k=2}^{n-1}\left ( \frac{\zeta-z_k}{1-\zeta z_k} \right )^{-\...
3
votes
0answers
94 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 ...
2
votes
0answers
27 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 ...
2
votes
0answers
44 views

Not sure if sparse quadrature routine is working correctly?

I have written a sparse quadrature routine to integrate multidimensional Gaussian integrals. I'm trying to show convergence plots in my thesis and demonstrate that the code works correctly. I am in ...
1
vote
0answers
35 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 ...
1
vote
0answers
23 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 ...
1
vote
0answers
37 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 ...
1
vote
0answers
226 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 $1$ ...
1
vote
0answers
105 views

solving numerically a 2D integral by using simps and quad combined

I have the following function $$f(x,y) =\left(\frac{1}{\exp(x-E_f)+1}-\frac{1}{\exp(x-y-E_f)+1} \right)\frac{1}{\sqrt{4-xy}}$$ I want to integrate it using Python 3. The domain of the $y$ variable ...
1
vote
0answers
75 views

Quadrature in finite element methods | How should I compute integrals involving the solution of the last time step?

Let $\Delta\subseteq\mathbb R^2$ denote the triangle spanned by $(0,0)$, $(1,0)$ and $(0,1)$ and $$\mathbb P_r(\Delta):=\left\{p:\Delta\to\mathbb R\mid p(x)=\sum_{|\alpha|\le r}\lambda_\alpha x^\alpha\...
1
vote
0answers
581 views

Numerical Double integration with endpoint singularity in scipy Python gives incorrect answer

I am trying to integrate the following function in Python, $\int_{0}^{\infty}\int_{0}^{\infty} \dfrac{e^{-x-y}}{B(x,y)}dx dy$, where $B(x,y)$ is the beta function - $B(x,y) = \int_0^{1}a^{x-1}(1-a)^{...
1
vote
0answers
9k 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 ...
1
vote
0answers
261 views

Numerical integral with a weakly singular kernel with a satisfactory precision

I am working on a numerical method for time fractional PDE. One problem is that I must compute a numerical integral of the following form: $$ \begin{equation} \int_0^{t_m} (t_m-s)^{-\beta}f(s)ds \end{...
0
votes
1answer
1k views

Faster code for double integration using Gauss-Legendre quadrature

I came with a the following code to evaluate a double integral using Gauss Legendre quadrature in MatLab ...