Also called numerical integration, quadrature refers to the approximation of an integral made by evaluating the integrand at a finite number of points.

116 questions
Filter by
Sorted by
Tagged with
74 views

### 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 ...
41 views

### 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 ...
150 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 ...
408 views

83 views

### 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, tau to z, tau. Since the integrand diverges at z=0.9, ...
45 views

### 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 ...
34 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 ...
22 views

### 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 ...
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 ...
80 views

### 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 ...
113 views

294 views

### Gaussian Numerical Differentiation

Gaussian quadrature improves on Newton-Cotes formulas by allowing the abscissas to vary along with the weights in order to integrate higher order polynomials. Can this idea be extended to numerical ...
301 views

234 views

### Numerical quadrature in Discontinuous Galerkin

I would like to know which is the best way to integrate numerically Legendre polynomials. I am building up a Discontinuous Galerkin CFD code for which Legendre polynomials are used as basis functions ...
215 views

### How to increase precision of Gauss-Legendre Quadrature in MAPLE? [closed]

I've written a simple legendre quadrature in MAPLE to compute the integral $\int_{-1}^1 r^2 dr = \frac{r^3}{3} |_{-1}^1=\frac{2}{3}$ ...
187 views

### Numerical integration of a hypergeometric function

The Task Let $z_1, z_2, z_3$ be positive real numbers and define $$r(\mathbf{z}):= \int_0 ^\infty (t+z_1)^{-3/2}(t+z_2)^{-3/2}(t+z_3)^{-1/2}\text{d}t.$$ The task is to compute $r$ numerically in ...
305 views

131 views

### Are there mesh generation methods that allow ugly elements?

Most mesh generation software seems to be aimed at building nicely shaped elements for FEM. I'm curious about a different situation: I need to numerically integrate over an irregular region. I don't ...
162 views

I need to numerically evaluate 2-D integrals of the form: $$\mathcal{I}(\theta) = \int_{0}^{1} \int_0^1 \varphi_\theta(x,y) dx dy$$ where $\varphi_\theta$ is a family of smooth functions indexed by ...
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 ...
I am doing a nested integral via quadrature. To give a definite example, lets say: $$\int_0^2dx \left[x + \int_0^x dy \, 2y\right]$$ So effectively I'm integrating $x + x^2$ from 0 to 2 (although ...