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Questions tagged [integration]

For questions related to integration on computers. This can include numerical approximations of integrals (e.g. Monte Carlo, quadrature, FEM, RK4) and algorithms/software to obtain analytical derivatives (Risch algorithm, SymPy).

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49 votes
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What does "symplectic" mean in reference to numerical integrators, and does SciPy's odeint use them?

In this comment I wrote: ...default SciPy integrator, which I'm assuming only uses symplectic methods. in which I am refering to SciPy's odeint, which uses ...
uhoh's user avatar
  • 1,058
13 votes
2 answers
8k views

Which Runge-Kutta method is more accurate: Dormand-Prince or Cash-Karp?

I simply want to know whether the Dormand-Prince Numerical Method or the Cash-Karp Numerical Method is more accurate.
ABBC's user avatar
  • 233
11 votes
3 answers
951 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}...
doetoe's user avatar
  • 593
11 votes
5 answers
9k views

Integral in log-log space

I'm working with functions which, in general, are much smoother and better behaved in log-log space --- so that's where I perform interpolation/extrapolation, etc, and that works very well. Is there ...
DilithiumMatrix's user avatar
11 votes
1 answer
226 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 ...
Arturo don Juan's user avatar
10 votes
3 answers
545 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 $...
Nico Schlömer's user avatar
10 votes
0 answers
342 views

Numerical integration using interval arithmetic, nowadays

Is there now a package for rigorous numerical integration that uses interval arithmetic and has access to a well-developed library of special functions? By "well-developed", I mean something that, at ...
H A Helfgott's user avatar
8 votes
2 answers
2k views

Energy conservation in RK4 integration scheme in C++

My colleague and I are trying to study the three-body problem, with different integration schemes, starting from the two-body problem. We implemented the symplectic Euler scheme and the Runge–Kutta ...
jack23456's user avatar
  • 171
8 votes
1 answer
203 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{...
jtravs's user avatar
  • 81
8 votes
1 answer
213 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 ...
Arturo don Juan's user avatar
8 votes
1 answer
253 views

When is it advantageous to iterate integrals numerically?

If there is an $(n+1)$-dimensional integral of the form $$ \int_{[0,1]^{n+1}} f(x, y)\,\mathrm{d}^n x \,\mathrm{d}y,$$ normally one would evaluate this using a multi-dimensional integration library ...
Kirill's user avatar
  • 11.4k
7 votes
3 answers
272 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) ...
Museful's user avatar
  • 255
7 votes
1 answer
178 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 ...
Rosh's user avatar
  • 73
6 votes
2 answers
965 views

Implementation of Monte-Carlo Integration

After reading the Wikipedia page for Monte-Carlo integration, I have understood the basic idea but I am having trouble implementing it for a general case. The integration that I am trying to do is $$ \...
Michael's user avatar
  • 71
6 votes
3 answers
271 views

Numerical integration giving incorrect sign

For my research, I need to integrate the following function: $$ W(z)=\int_0^{\infty}dx\ w(x,z)\\ =\int_0^{\infty}dx\frac{e^x}{(e^x+1)^2}\log{\left(\frac{e^{z^2/4x+x}+1}{e^{z^2/4x+x}-e^x}\right)}\\ =\...
surrutiaquir's user avatar
6 votes
3 answers
872 views

Numerical integration using RKF7(8) - different results

For my thesis, I look in trajectories of vehicles through an atmosphere at very high velocities. I have a set of equations of motion, which I propagate using a Runge-Kutta-Fehlberg (RKF) 7(8) ...
theflyingaviator's user avatar
6 votes
1 answer
443 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(\...
HerpDerpington's user avatar
6 votes
1 answer
267 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 ...
Eric Kightley's user avatar
6 votes
0 answers
98 views

What is the best method to do a MC Integration of a multidimensional integral where the integration limits depend upon other variables?

What is the best method to do a Monte Carlo Integration of a multidimensional integral where the integration limits depend upon other variables? I am interested in getting a numerical value of a 5 ...
pollux33's user avatar
5 votes
2 answers
3k views

CUDA & Python for numerical integration and solving differential equations

Can anyone please suggest some libraries which allow use CUDA in Python for numerical integration and/or solving of differential equations? My goal is to solve large (~1000 equations) of coupled non-...
Artem Alexandrov's user avatar
5 votes
3 answers
5k views

Integration of the Fermi distribution using Python

I want to calculate the carrier concentration of my semiconductor using this equation: $$ n(x) = \frac{m^*}{\pi\hbar^2}\int_{E_k}^{\infty}\frac{1}{1+\exp\left(\frac{E-E_f}{k_BT}\right)} \mathrm{d}E $$...
ju.'s user avatar
  • 153
5 votes
1 answer
1k views

Why is velocity Verlet better than Verlet for gravity if it has a worse order of magnitude for the error term

Even though this method is more widely used than the simple Verlet method mentioned above, it unfortunately has an error term of O(Δt^2) , which is two orders of magnitude worse. That said, if you ...
ght007's user avatar
  • 95
5 votes
2 answers
3k views

Evaluating the surface integral in an FEM (Finite Elements Method) procedure

I want to evaluate the Surface force integral in an FEM procedure. The basic reference tet is shown in the figure. The faces are numbered corresponding to the node opposite to them. For example the ...
kaush's user avatar
  • 107
5 votes
1 answer
348 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 ...
e-eight's user avatar
  • 163
5 votes
1 answer
2k views

2D numerical integration with infinite limit (C++)

In order to integrate a two dimensional function of the form $$\int_{1}^\infty \int_{-\sqrt{x^2-1}}^{\sqrt{x^2-1}} e^{-x} \rm{d}y \rm{d}x,$$ I have been attempting to use the following code (written ...
user146268's user avatar
5 votes
1 answer
446 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 ...
Adr's user avatar
  • 173
5 votes
1 answer
1k views

Convergence of Monte Carlo integration

In my research, one of the steps is to choose a numerical method to estimate $\int_a^b f(t)dt$, where $f$ is Lipschitz continuous but not differentiable. For simplicity, I used midpoint rule but the ...
John's user avatar
  • 255
5 votes
0 answers
513 views

Fast integration scheme for path integral of Gaussian over a cubic curve

I need to numerically compute an integral of the following form: $$\int_0^1 \frac{1}{2\pi\sigma^2}\exp\left(-\frac{\|(q_0t^3 + q_1t^2 + q_2t + q_3) - a\|^2}{2\sigma^2}\right)\|3q_0t^2 + 2q_1t + q_2\|\,...
Phil's user avatar
  • 151
5 votes
0 answers
180 views

Why does this integral converge faster than normal rectangle or trapz integration?

I was looking for the fastest converging method to integrate a family of functions. After some tries, an old-school colleague suggested me a method that he used to use in excel to perform such task. ...
Luca's user avatar
  • 151
5 votes
0 answers
181 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 ...
interoception's user avatar
5 votes
0 answers
87 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 ...
interoception's user avatar
4 votes
3 answers
1k views

How can I numerically integrate the Kepler problem?

I tried to solve a simple Kepler problem numerically. I have discrete time steps, a starting position $(x_0,y_0)$ and starting velocity $(u_0, v_0)$. I used this iteration by calculating the forces ...
MichaelW's user avatar
  • 151
4 votes
4 answers
1k 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,...
Vazuoeow's user avatar
4 votes
1 answer
126 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 ...
Astrom's user avatar
  • 143
4 votes
1 answer
305 views

How to solve this set of equations involving an integral?

I have the following set of equations: $$ x(t) = x_0 \psi, \qquad y(t) = \kappa \ln \psi - x_0 \psi +1,\qquad z(t) =-\kappa\ln \psi,$$ with $$ t- t_0 = \int ^\psi_{\psi_0} \frac{d\eta}{\eta(1+\...
AngusTheMan's user avatar
4 votes
1 answer
429 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}...
dba's user avatar
  • 295
4 votes
1 answer
1k 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. ...
Galilean's user avatar
  • 151
4 votes
0 answers
144 views

Quadrature rules for non-linear finite element problems

For solving linear problems stemming from PDEs with the FEM, such as the Poisson equation or the wave equation, it is customary to use the "simplest" numerical quadrature that exactly ...
Andreas Longva's user avatar
4 votes
0 answers
98 views

Sample Average Approximation vs. Numerical Integration

To calculate the expected value of objective functions, we have two choices: Sample Average Approximation (SAA): $$ \frac{1}{N}\sum_{i=1}^N f(x,\xi^i). $$ Numerical Integration (e.g., Monte Carlo ...
Keith's user avatar
  • 41
4 votes
0 answers
304 views

How to numerically evaluate this double Integral?

I want to evaluate the following integral: $$\int_{0}^{60} \ \left(\int_{0}^{2z} 0.5\cdot t \left(\mathrm{erf}(t-a) -1 \right)J_{0}(qt)\mathrm{d}t \right)^2 \mathrm{exp}\left(-\frac{(z-a)^2}{2s^2}\...
Shankar_Dutt's user avatar
4 votes
0 answers
261 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 - ...
Tony_V's user avatar
  • 73
4 votes
0 answers
90 views

Can I make this numerical integration continuously differentiable?

Suppose I have the discrete values $f(x_i)$ for every discrete value $x_i$ greater than some $\varepsilon$, and I want to numerically calculate the following integral: \begin{equation} n = \int_\...
Maziar Noei's user avatar
3 votes
4 answers
2k 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 ...
curious_cosmo's user avatar
3 votes
2 answers
447 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 ...
ElectronsAndStuff's user avatar
3 votes
1 answer
17k views

Composite simpson's rule with odd intervals

The composite simpson's rule subdivides the interval into n equal subintervals, with n even. Then $$\int_a^b f(x) dx \approx \frac{h}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 2f(x_{n-2}) + 4f(...
bernie's user avatar
  • 165
3 votes
1 answer
168 views

Optimal quadrature rule for heavy tail measure

I'm looking for a well-thought quadrature rule for this measure $d\mu(t)=\frac{dt}{t^s}$ for $s\in(0,1)$, the underlying motivation is to compute this integral $$ \lambda^{s-1}=\frac{1}{\Gamma(1-s)}\...
Aner's user avatar
  • 151
3 votes
2 answers
2k views

Inaccurate results of integration using scipy solve_ivp

I am trying to use solve_ivp to solve the following 1st order ODE: $$ \frac{d \rho}{d z} = \frac{m \theta}{(1+\theta z)} \, \rho, $$ subject to $\rho(z=0)=1$, where ...
Fryderyk's user avatar
3 votes
1 answer
90 views

Hankel transform with high accuracy

Short version I'm computing the zero-order Hankel transform of a function $f(r)$, $$F(k) = \int_0^\infty f(r)J_0(kr)r\,\mathrm{d}r$$ I know $f(r_n)$ at selected $r_n$ but it is impractical to compute ...
David Z's user avatar
  • 3,393
3 votes
1 answer
649 views

General heuristics for making a choice "dopri5", and "lsoda"?

With scipy, I have the choice of using "lsoda": Real-valued Variable-coefficient Ordinary Differential Equation solver, with fixed-leading-coefficient ...
bzm3r's user avatar
  • 659
3 votes
1 answer
206 views

Question about energy in the shallow water equations on a staggered grid

I think this is a question about the energy conservation of a numerical integrator. I'm studying the linearized 1D shallow water equations in python - for reference, here they are: $$ \frac{\partial u}...
theWrongAlice's user avatar

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