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).
216 questions
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Quadrature of rational functions
I have a class of integrals I need to solve numerically which have the form:
$$
I_k = \int_a^b \frac{p_k(x)}{x^k} dx, \quad k = 0, 1, \dots, K
$$
where $p_k(x)$ is a cubic polynomial on the interval $[...
- 1,078
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1
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How to deal with solving coupled ODE systems where variables are updated multiple times within each timestep?
I'm solving a system of coupled ODEs using Euler integration for simplicity. To make this concrete, please see the (extremely simplified) minimal working example below in Python. Imagine we have a box ...
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2
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1
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High precision numerical integration of discrete data with Matlab
I have discrete data of a function plotted below:
The "Y" values of the function near "X=1.57" are very close to each other and zero, like 9.25558265263186E-11 and 5....
- 23
4
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0
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159
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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 ...
2
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0
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518
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Efficient way of calculating a cumulative integral with prefactor
I have a grid of points $x_i$ and corresponding function values $y_i=f(x_i)$. I'm interesting in something like the cumulant of $f$, but it has an awkward prefactor. The desired quantity we'll call
$$...
3
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1
answer
231
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Question about energy in the shallow water equations on a staggered grid
I think this is a question about the energy conservation of a numerical integrator.
I'm studying the linearized 1D shallow water equations in python - for reference, here they are:
$$
\frac{\partial u}...
- 105
2
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0
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514
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Numerical evaluation of Duhamel's integration
I am trying to numerically evaluate the following Duhamel's integration:
$$
x = \frac{-1}{\omega_d} \int_0^t \ddot{x}_g (\tau) e^{-\zeta \omega_n(t - \tau)} \sin{\left( \omega_d (t - \tau) \right)} d\...
- 475
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3
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Numerical integration giving incorrect sign
For my research, I need to integrate the following function:
$$
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)}\\
=\...
- 105
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0
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76
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Fourier integral over elements
Suppose I have a triangular element with vertices ${\vec{r_1},...,\vec{r_3}}$ and a function $f(\vec{r})$. I want to calculate the fourier integral over this triangle such that:
$$F(k_x,k_y)=\int \int ...
- 11
3
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1
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Importance Sampling for multidimensional integrals and random numbers from multivariable pdf's
I am aiming to get a numerical value for a five-dimensional integral using Monte Carlo Integration. I am getting good results using the Mean Value Method, but I would like to try to use Importance ...
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0
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What is the best method to do a MC Integration of a multidimensional integral where the integration limits depend upon other variables?
What is the best method to do a Monte Carlo Integration of a multidimensional integral where the integration limits depend upon other variables?
I am interested in getting a numerical value of a 5 ...
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1
vote
1
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459
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Compute 2D numerical double integration with Boost C++ with parameters
I am trying to compute the double Richardson and Wolf integrals for the focusing of a lens with Boost in C++ (using the Gauss Kronrod method).
As a starting point, I used the example presented in this ...
4
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103
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Sample Average Approximation vs. Numerical Integration
To calculate the expected value of objective functions, we have two choices:
Sample Average Approximation (SAA):
$$
\frac{1}{N}\sum_{i=1}^N f(x,\xi^i).
$$
Numerical Integration (e.g., Monte Carlo ...
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2
votes
1
answer
6k
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Error in Simpson's 3/8 rule is higher than that of Simpson's 1/3 rule
For a given function $f(x)$, I have tried to find its numerical integral using Simpson's 1/3 and Simpson's 3/8 rules.
I then compare the solution from the numerical quadratures to the analytical ...
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1
vote
1
answer
84
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Definite Numerical Integration with Unknown limit
How to solve for small gamma in the integral equation in Scipy ? I recognize it has to be solved with both the numerical integral and a root solver (Newton's method)
$$ \int_{\gamma}^{+\infty}f(x) dx =...
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1
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Does the leap-frog algorithm conserve energy for n-body problems?
The leap-frog algorithm is able to conserve to a certain extent the energy of a system, which flucutates as a cosine around a stable value. Is this true if we apply the algorithm to a n-body ...
-1
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1
answer
207
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Solving ODEs, Rotations, Angular Velocity, Euler Angles
I am implementing a simulation that needs to rotate and object based
on known angular velocity (assumed constant for simplicity). I followed the
ideas given below, pg. 32)
https://graphics.stanford....
- 239
5
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0
answers
557
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Fast integration scheme for path integral of Gaussian over a cubic curve
I need to numerically compute an integral of the following form:
$$\int_0^1 \frac{1}{2\pi\sigma^2}\exp\left(-\frac{\|(q_0t^3 + q_1t^2 + q_2t + q_3) - a\|^2}{2\sigma^2}\right)\|3q_0t^2 + 2q_1t + q_2\|\,...
- 151
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2
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Computing infinite series with iterated functions
I found this question (linked here) which asks to find what this infinite series converges to
$$ \sum_{n=1}^{\infty} \int_0^{\pi} f_n(x) dx $$
where $f_{n+1}(x) = \sin(f_n(x)) $ and $f_1 = \sin(x)$. ...
- 111
2
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3
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317
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Calculating the antiderivative numerically
In the standard literature topics like numerical differentiation and numerical integration are usually discussed in detail. However, numerical integration is not the same as calculating the (true) ...
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Free Electron Schrödinger Equation (Energy Method)
For the simplest atom, its wave function is described by the PDE of Schrodinger equation:
$$
-i h \frac{\partial u}{\partial t }=\frac{h^{2}}{2m} \Delta u + \frac{e^{2}}{r}u$$
The potential $\frac{e^{...
user38211
1
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1
answer
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Numerical integrator for $a'(t)=e^{-a(t)}f(t)$
Suppose I know a function $f(t)$ and all its derivatives in $t$ in closed form. Given $a(0)$ and some $t_0>0$, I'm looking for an explicit integrator that can estimate $a(t_0)$, where $a(\cdot)$ ...
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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.
...
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2
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0
answers
736
views
Errors in Integral Estimate of Gaussian using Trapezoidal Rule
I'm trying to estimate the percentage error in computing the integral of a Gaussian via composite trapezoidal rule versus via an exact formula. To do this I've generated a gaussian with mean 0, ...
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1
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573
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ODE45 and a variable that assumes multiple values during the timespan
I have tried in different ways to see what happens to voltage V and gating conductances m, n and h when, at time step x, current I switched from 0 to 0.1, and then at time step x + n it gets back to 0....
- 1
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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-...
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2
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226
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Integrate a function from samples using computer codes
I have a function $c ( I (\vec{r}) )$. Not a constant, $c$ doesn't denote a constant. So $c$ is a function of $I$ which is a function of $\vec{r}$. $I$ is an intensity (W/cm2).
This $c$ is hard to ...
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2
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1
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998
views
nquad Integration in SciPy
I am trying to self-learn SciPy and evaluate the following quadruple integral using scipy.integrate.nquad:
$$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1-x} (...
- 125
1
vote
1
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73
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Gauss Integration of $\sqrt(x)$
I want to construct a gauss integration for the weight function $w(x) = x^{1/2}$ for
$$\int_{0}^{1}x^{1/2}f(x)dx = a_{1}f(x_{1})+a_{2}f(x_{2})$$
Solving
\begin{align*}
a_{1}+a_{2} =& \int_{0}^{...
user37062
2
votes
1
answer
2k
views
Trouble with backwards time integration in Python
I am struggling with a rather basic numerical integration task: Using Python's scipy.integrate.solve_ivp module to integrate an ODE sytem backwards in time. As a test, I am using the following ODE ...
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0
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How to derive the adjoint sensitivity equations for a least squares objective function gradient
The Problem
I would like to determine the gradient of a least squares objective function which depends on a vector of 40 parameters $p$, and the solution of a system of 32 differential equations. In ...
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0
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94
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Integration of a discretized field in a cylindrical coordinate system
I would like to integrate a discretized field in cylindrical coordinates, given as A(r, z), with z being spaced regularly (...
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1
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Trouble Making 3rd-Order Sympletic Integrator for Planitary N-Body Problem (A Hamiltonian System)
I am doing a solar-system simulation. I am using Ruth's 3rd order sympletic integrator to avoid the problem of Energy Drift (which I had with RK4), but the the planets quickly leave orbit, and energy ...
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2
votes
1
answer
3k
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Numerical integration problem: IntegrationWarning The integral is probably divergent, or slowly convergent
I am trying to get the numerical integration of a function using scipy's integrate.quad as follows.
$$
\begin{equation}
G (\alpha) = \frac{4\alpha}{\pi}\int_0^{\...
- 23
3
votes
1
answer
475
views
Choosing an appropriate time step for a discrete & continuous dynamics simulation
I have inherited of a flight dynamics simulation in C++ which represents a small drone with it's autopilot, actuator dynamics and a solid state IMU.
Hence, it is composed of a few models, some ...
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3
votes
1
answer
323
views
Find quadrature points and weights
I'm struggling with the following problem:
What is the maximum degree of exactness that we can obtain with the following quadrature >formula
$$\int_0^1 f(x)\frac{1}{\sqrt{x}}dx \approx w_0 f(x_0) +...
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0
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109
views
Nondifferentiable coordinate transforms
Suppose that we have coordinates $u=u(x,y)$ and $v=v(x,y)$ in $\mathbb{R}^2$ so that $v$ is not differentiable when $u(x,y)=u_0$ where $u_0$ is a constant. Can we solve a differential equation, such ...
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How to use numerical integration to calculate the surface area of a superellipsoid?
I am working in an application in which I need to calculate the surface area of a superellipsoid. I have read that there is no closed form solution (see here), so I am trying to compute it using ...
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vote
0
answers
77
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Computation of a functional for large values
Consider the following function :
$$f(x) = \sin^2(\frac{π\Gamma(x)}{2x})$$
Now consider the following functional :
$$I(x)=\int_0^\infty \frac{f(x + iy) − f(x − iy)}{e^{2πy}-1} dy$$
I need values for ...
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2
votes
2
answers
498
views
Weak form of the Navier-Cauchy equation
I am trying to obtain the weak form of the Navier-Cauchy equation, which is
$$- \rho \omega ^2 \textbf{U} - \mu \nabla ^2 \textbf{U} - (\mu + \lambda) \nabla (\nabla \cdot \textbf{U}) = \textbf{F}$$
...
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0
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views
Is Romberg integration method implemented as weighted function values numerically correct?
I have to integrate expression f(x) * g(x) for many different functions f but just one g.
I ...
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votes
2
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265
views
Integration by parts with cross derivatives to obtain the weak form [duplicate]
I’m trying to write the weak form of the Navier-Cauchy equation in the component form, where $u_1$ and $u_2$ are the displacement components:
$$-(2 \mu +\lambda) \frac{\partial ^2 u_1}{\partial x_1 ^2}...
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1
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How to use cumtrapz correctly?
I have tried to do a trapeze integration with f(x)=x^2, where I know how the antiderivative looks like, so F(x) = (1/3)x^3
Here's my code, just like I tried:
...
0
votes
1
answer
256
views
Two RK4 method in one program
I want to solve this integral using RK4 by coding in Fortran:
$$R=∫1/a(t) dt → dR/dt=1/a(t) =f(t)$$
Initial point: t=0 (or a=0.001) and R=0
And I have to get a(t) by solving another ...
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votes
0
answers
53
views
To use the confluent hypergeometric function or not to?
I am numerically computing the following integral as a function of positive $k$.
$$I(k) := \int_0^\infty x^b(k+x)^{a-1} e^{-x} dx \tag1$$
It is shown in math.stackexchange.com that this can be ...
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3
votes
1
answer
206
views
Time independent Runge Kutta integration of SDE
I am trying to compare the result of numerical integration of time independent Runge_Kutta, github page for stochastic differential equations with the analytical solution.
True answer match the ...
- 131
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votes
0
answers
27
views
Does order of data points matter for approximating AUC with unit time steps?
I have a time series of data where the increment is every minute. In order to approximate AUC, I just compute the sum of the data values, since they would all be multiplied by 1 anyway per Riemann sum ...
pyhat32
1
vote
0
answers
89
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Plot of ratio of two integrals:
Consider the following integrals
$$
I_1(x) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy) − F(x −\mathrm iy)}{\mathrm e^{2πy}-1},
$$
And
$$I_2(x) =\int_1^x F(t)dt$$
Where, $ F(z) = \sin^2[π\Gamma(z)/...
- 119
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votes
0
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339
views
How to numerically evaluate this double Integral?
I want to evaluate the following integral:
$$\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}\...
- 61
1
vote
3
answers
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views
Comparison of integrals with a function:
Consider the following integral:
$$S(q)=\int_{x=2}^q\sin^2\left(\frac{π\Gamma(x)}{2x}\right)dx$$
And consider the functions :
$$R(q)=\frac{q}{\log(q)}$$
$$T(q)=\int_2^q\frac{1}{\log(x)}dx$$
I ...
- 119