Questions tagged [integration]

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16 views

Double Integral with Gauss- Hermite for one component

I am trying to perform the following integral $$\iint_V \frac{r'\left(e^{-r'^2/2\sigma^2}\right)\left(r-r'\cos(\theta-\theta')\right)}{r^2+r'^2-2rr'\cos(\theta-\theta')}dr'dθ'$$ Using Gauss-Hermite ...
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1answer
40 views

Plot of a function involving an integral and value changing parameters [closed]

I'm trying to plot the cross section with respect to the photon energy $h\nu$ but for $\gamma = 1.0, 1.2, 2.0 $ in the same axes $\sigma = \left[\left(\frac{\xi_{eff}}{\xi_{0}}\right)^2 \frac{n_r}{\...
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1answer
39 views

How do I find the portion of a cell/voxel lying within a defined surface?

We have a 3-dimensional grid of voxels (or cells), with individual voxels being of volume $dx\,dy\,dz$ where $dx=dy=dz=1$. A cone-like surface is defined by some function, $z = f(x, y)$, which in ...
2
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1answer
29 views

Adaptive Runge-Kutta for Stochastic (Projected) Gross-Pitaevskii Equation

I am using the XMDS library for solving the stochastic (projected) Gross-Pitaevskii equation $$i \hbar \partial \Phi\left(\mathbf{r},t\right)_t=\hat{\mathcal{P}}\left\{(1-i \gamma)\left(\hat{H}_{\...
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0answers
45 views

Solve the 1D Poisson equation by integrate twice the charge density

Let's say I have a set of data (1-D array) called charge density along the $z$ direction (obtained from DFT calculation), and I want to integrate it twice with respect to $z$ coordinates points (i.e., ...
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0answers
31 views

How to integrate using the composite trapezoidal rule without numpy/scipy [duplicate]

For this I am using Python. I need to use the composite rule of Boole to integrate a function. I know there is a function in numpy (numpy.trapz) that does the integration, but I haven't found one for ...
2
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1answer
71 views

Using Implicit Euler with second order differential equations

We can numerically integrate first order differential equations using Euler method like this: $$y_{n+1} = y_n + hf(t_n, y_n)$$ And with Implicit Euler like this: $$y_{n+1} = y_n + hf(t_{n+1},y _{n+...
2
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0answers
39 views

Evaluating integral $F(r_2) = \frac{1}{r_2^n} \int_0^{r_2} r_1^n f(r_1)\ \mathrm{d}r_1$ without growing instability

I have the following expression to be numerically integrated in a vector-based library (e.g. numpy, MATLAB, etc), $$ F(r_2) = \frac{1}{r_2^n} \int_0^{r_2} r_1^n f(r_1)\ \mathrm{d}r_1, $$ where $n$ is ...
4
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4answers
412 views

Trapezoid rule vs Gaussian quadrature: what am I missing?

I'm reading a paper right now which criticizes a method because it uses trapezoid rule, rather than "more advanced quadrature rules like the Gauss quadrature"... The Gaussian quadrature rule requires,...
3
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2answers
180 views

Solving ODE with “Jumpy” Coefficients

I'm numerically solving a linear coupled ODE of the form $$y^{\prime}(t) = \hat{M}(t)y(t)=\left[\begin{array}{cc}0& A(t)\\ B(t)& 0\end{array}\right]y(t),$$ and the difficulty I'm running into ...
2
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0answers
26 views

2-dimensional Gauss-Hermite quadrature in R

A similar question was asked here and the given answer is perfect for a unidimensional integration. I need to make bidimensional integration in R with a Gauss-Hermite quadrature: $$\int_{R^2} h(p1,p2)...
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0answers
53 views

Is it possble to do this complex symbolic calculation with Matlab?

Sorry it's bit abrupt, but recently I am caught up in some symbolic calcualtion which is tedious and almost impossible with mere human hands, so just wondering is it possible to solve the double ...
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2answers
85 views

Solving a 1D diffusion equation with linear and nonlinear source terms

I would like to numerically solve the following equation: $$\frac{\partial \rho (z,t)}{\partial t} = B(N_D \rho (z,t) + \rho(z,t)^2) + D \frac{\partial^2 \rho (z,t)}{\partial z^2}$$ with the boundary ...
3
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0answers
67 views

Numerical integration with singularity term

In https://www.johndcook.com/blog/2012/02/21/care-and-treatment-of-singularities, the author explains the subtraction method to get rid of singularities when performing numerical integration. The ...
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0answers
36 views

FFT convolution works only with certain domain length

in my quest to understand how I can use FFT to compute integrals (see my other question click, still no answer there), I came across the fact that a convolution of two functions can be calculated by ...
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2answers
80 views

Is expm1 the right primitive?

I'm writing some code to calculate $\int_0^1 e^{ax} \mathrm{d} x$. Annoyingly there does not seem to be a way of doing this without if statements: ...
2
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1answer
45 views

Numerical integration of the dataset of a function

The energy equation for a spherically symmetric system is given by $$\mathscr{E}=\frac{v^2(r)}{2}+\frac{c_s^2(r)}{\gamma-1}+\phi(r)$$ where $\mathscr{E}$ is the total energy, $v$ is the velocity of ...
1
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1answer
76 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 ...
1
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2answers
50 views

Numerical integral with symbolic integral in exponent

Many times in fourier approximation we come across integrals such as $$\int_0^1 e^{-\gamma\int_0^xu_0(\eta)d\eta}dx$$ where $\gamma$ is a constant and the data for $u_0$ is provided as a discretely ...
2
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0answers
37 views

Monte Carlo domain not-so-dense

I already posted it on Physics SE, but maybe this is a better place: I have a 5D integral being calculated with a Monte Carlo uniform random sampling. The issue is that the region of integration is ...
7
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1answer
131 views

Numerically estimating expected value of f(x) when x is normally distributed

I need to estimate $$ \mathbb{E}_x[f_i(x)] = \int_{\mathbb{R}^n} f_i(x) p(x) dx $$ for many functions $f_i(x)$, where $p(x)$ is the density of a normal distribution. The evaluation of all the ...
4
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1answer
224 views

DG-FEM integration by parts

I am going through the book of Hesthaven and Warburton on discontinuous Galerkin methods. I have difficulties understanding some basic steps in the calculations. Consider the PDE: $$\frac{\partial u}...
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0answers
38 views

Techniques to optimise the integral of a function of known analytical form

I need to compute repeatedly a function that depends on an integral. The integral is not solvable analytically, but it depends on the argument of the function parametrically, like this: $$ f(x) = \...
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0answers
31 views

Execution time of cumulative integral

In Matlab, we have the cumtrapz function, which returns the approximate cumulative integral of y: I = cumtrapz(x,y) This ...
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3answers
317 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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1answer
84 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[0], tau[0] to z[2249], tau[2249]. Since the integrand diverges at z=0.9, ...
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0answers
60 views

Quadrupole moment for a right triangle

The authors define a quadrupole moment for a right triangle in Lazić, Predrag, Hrvoje Štefančić, and Hrvoje Abraham. “The Robin Hood Method – A Novel Numerical Method for Electrostatic Problems ...
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1answer
94 views

What's the minimum step size that can be used in Euler's method before it becomes unreliable?

In particular, if Euler's method is implemented on a computer, what's the minimum step size that can be used before rounding errors cause the Euler approximations to become completely unreliable? I ...
3
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4answers
355 views

Numerical integration in Python with unknown constant

I’d like to solve the below equation for the unknown $T$: $$\int_0^\infty \frac{x^2}{\exp\left(\frac{x}{T}\right)-1}\kappa_x \mathrm{d}x = C,$$ where $C$ is a known constant and $\kappa_x$ is some ...
4
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0answers
112 views

MATLAB: Compute the Schwarz-Christoffel transformation symbolically

Suppose we have a conformal mapping from the unit disk in the $\omega$ plane onto the exterior of a polygon in the $z$ plane. The Schwarz-Christoffel mapping in this case is defined as: $$f(u) = A - ...
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1answer
101 views

calculating integral for an ODE system

I have an ODE system defining a mathematical model of a biological system, say $$ \frac{da}{dt}=f_1(a,b,\ldots,z,p)\\ \frac{db}{dt}=f_2(a,b,\ldots,z,p)\\ \cdots\\ \frac{dz}{dt}=f_n(a,b,\ldots,z,p) $$ ...
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0answers
109 views

Best way to numerically compute elliptic integrals of the third kind with complex arguments?

I need to compute elliptic integrals of the third kind with complex arguments, preferably in C++. Is there code out there to do this? I have discovered the Arb library, but that does much more than I ...
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3answers
76 views

Rudin lecture — if f(x) is not integrable on some interval, does it not have a Fourier Series expansion on that interval?

I found an old lecture on YouTube given by Walter Rudin (1990, in Wisconsin), and towards the beginning he mentions that if $f(x)$ were not integrable, on some interval, it would be obvious that it ...
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0answers
76 views

Is there a numerically stable way to take $\epsilon \rightarrow 0$ in integrals of the form $\int \frac{f(x)dx}{x+i\epsilon}$?

The Sokhotski-Plemelj theorem states, $$\lim_{\epsilon\rightarrow 0^+}\int_a^b\frac{f(x)dx}{x+i\epsilon} = \mathcal P \int_a^b \frac{f(x)dx}{x} - i\pi f(0). $$ Is there a numerically stable way to ...
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1answer
113 views

How do I integrate a function defined over an arbitrary area?

Let's say, I have a compact area $S$ (for example a circle, a square or some arbitrary polygon) and a function $f: S \rightarrow \mathbb{R}$. I want to numerically calculate the Integral $$ \int_S f(\...
0
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1answer
32 views

FEM 1D poisson substitution integral issue

I'm trying to solve $ \begin{cases} -u''=f \\ u(0)=0 \\ u(1)= \alpha \end{cases} $ with FEM using reference elements and local coordinates. So we have the global ...
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2answers
198 views

How many quadrature points should I use?

I am trying to compute the following integration $$ \int_0^\infty e^{-y}y^{a/2}L_c^b(y)L_e^d(y)dy $$ using the generalized Gauss-Laguerre quadrature routine in the GNU Scientific Library. Here the $L$'...
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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 ...
4
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1answer
134 views

What is the best numerical method for a six dimensional spherical integral?

I am trying to do integrals of the type $$ \int d^3\vec{p} \int d^3\vec{p}' e^{-p^2} e^{-{p'}^2}f(\vec{p}, \vec{p}') $$ where $\vec{p}$ and $\vec{p}'$ are three dimensional vectors represented using ...
3
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2answers
916 views

Integration of the Fermi distribution using Python

I want to calculate the carrier concentration of my semiconductor using this equation: $$ n(x) = \frac{m^*}{\pi\hbar^2}\int_{E_k}^{\infty}\frac{1}{1+\exp\left(\frac{E-E_f}{k_BT}\right)} \mathrm{d}E $$...
8
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1answer
166 views

What are these oscillations?

I have a function $g(x)$ defined numerically that is sort of in between a Gaussian and a Lorentzian. It decays much slower than a Gaussian, but still faster than a simple inverse power. I need to ...
11
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1answer
173 views

Numerically Recovering Imaginary Part of Analytic Continuation from Real Part

My situation. I have a function of a complex variable $f(z)$ defined through a complicated integral. What I am interested in is the value of this function on the imaginary axis. I have numerical ...
2
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1answer
93 views

Two variables integration matlab

I'm trying to solve physical problem in quantum mechanics of helium atoms, the solution require numerical integration over 2 variables. However when i'm trying to run the next code ...
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0answers
47 views

Book Suggestion for Approximating Integrals using Random Partitions

Suppose I want to approximate the integral $\int_0^1 x^2\,dx$ using Riemann Sums or Darboux sums over random partitions of the interval $[0,1]$, Like in the image below: Here, A "random" partition of ...
2
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2answers
297 views

Integration of a diverge function in c++ GSL Library

I am trying to perform an Integral of Hypergeometric function 2F1(a,b,c,x) from -1 to 1 for some good values of $a,b,c$ (lets say $a=1,b=2,c=3$) . I did it in ...
7
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3answers
143 views

Evaluating an integral numerically at many points

Given a real function $f$, how can one efficiently evaluate $\int_0^{a_i}f(x)dx$ for millions of different $a_i$? Applying a standard quadrature method (such as Simpson's rule or Gaussian quadrature) ...
4
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1answer
80 views

Error on a integral quantity with noise

First of all sorry if this is the wrong place to ask this question, I went to a few stack sites and thought here it would be more suitable. My problem: I have a physical quantity $F$ that depends on ...
1
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1answer
156 views

Scipy odeint Unexpected Results

I am attempting to numerically integrate the equation $$\frac{\mathrm{dP} }{\mathrm{d} r}=-\left ( P+\rho\left ( r \right ) \right )\frac{m\left ( r \right )+4\pi r^{3}P}{r\left [ r-2m\left ( r \...
1
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1answer
108 views

Weighted Monte Carlo Integration

I have a function $F(x)$ which drops exponentially (like differential QCD cross section vs. Invariant mass). I want to perform Monte-Carlo integration. The problem is that only small $x$'s which have ...
2
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0answers
111 views

Why would someone use empirical sum instead of numerical integration methods?

In the context of a scientific computing application, using data coming from (powerful) embedded systems, acquiring raw data (but from calibrated acquisition electronics), I have been asked to ...