Questions tagged [integration]

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2
votes
3answers
162 views

Calculating the antiderivative numerically

HeIIo everybody, in the standard literature topics like numerical differentiation and numerical integration are usually discussed in detail. However, numerical integration is not the same as ...
2
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0answers
61 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, ...
0
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1answer
81 views

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^{...
-1
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0answers
59 views

Scipy quadrature integration inaccuracy

I have Python code which evaluates the complex integral of a single variable function. My definition of integration is as follows: ...
2
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1answer
130 views

Double Integral with Gauss- Hermite for one component

I am trying to perform the following integral $$\int_{0}^{2\pi}\int_{0}^{+\infty} \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'...
-1
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0answers
32 views

Spliting multiple variable C++ functions for quadrature integration [migrated]

I want to numerically integrate with boost::math::quadrature::trapezoidal(g, a, b, 1e-6); Here I'm integrating the function g(x). The problem is that I have to ...
1
vote
1answer
102 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 ...
0
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1answer
82 views

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 $a(\cdot)$ ...
5
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0answers
121 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. ...
0
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2answers
75 views

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 ...
0
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0answers
25 views

How to efficiently perform this 2D integral in Quadpy?

I need to integrate a function defined in 2Dims (z and radius r), for which I don't have an expression. I can just query the ...
-1
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1answer
60 views

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....
3
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2answers
293 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-...
3
votes
1answer
67 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 ...
1
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1answer
98 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} (...
1
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1answer
67 views

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}^{...
0
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0answers
10 views

Different values of Gaussian Integration in Pracma library in R

I want to construct a gauss integration for the weight function: $$w(x) = x^{1/2}$$ $\in(0,1)$ of the form: $$\int_{0}^{1}x^{1/2}f(x)dx = w_{1}f(x_{1})+w_{2}f(x_{2})$$ \begin{align*} w_{1}+w_{2} =...
1
vote
1answer
184 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 ...
2
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0answers
84 views

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 ...
0
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0answers
31 views

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 (...
1
vote
1answer
106 views

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 ...
1
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1answer
415 views

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^{\...
2
votes
2answers
211 views

Weak form of the Navier-Cauchy equation

I am trying to obtain the weak form of the Navier-Cauchy equation, which is $$- \rho \omega ^2 \textbf{U} - \mu \nabla ^2 \textbf{U} - (\mu + \lambda) \nabla (\nabla \cdot \textbf{U}) = \textbf{F}$$ ...
3
votes
1answer
67 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 ...
2
votes
1answer
80 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) +...
0
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0answers
92 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 ...
2
votes
2answers
309 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
votes
1answer
206 views

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 ...
34
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2answers
6k views

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 ...
1
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0answers
72 views

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 ...
1
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0answers
99 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 ...
1
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0answers
76 views

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)/...
0
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2answers
165 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}...
0
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1answer
168 views

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
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1answer
118 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 ...
2
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0answers
38 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
3answers
263 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 ...
0
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0answers
21 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 ...
5
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0answers
118 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}\...
2
votes
1answer
253 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. ...
2
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0answers
90 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)...
0
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3answers
85 views

Changing variables in integral to avoid infinity

I want to write a code in Fortran to solve this integral numerically: $$\int_{1095}^\infty \frac{dx}{x\sqrt{(x+644.153)(4.17 \cdot 10^{-5} x+0.145)}}$$ What is the best method for it? I tried Monte-...
2
votes
1answer
651 views

Vectorised second order ode solving in python

I am trying to write a python program that simulates the motion of a large number of particles by numerically integrating a second order ordinary differential equation. I first split the ODE into two ...
-1
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1answer
105 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}{\...
2
votes
1answer
57 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}_{\...
2
votes
1answer
108 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
40 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 ...
3
votes
4answers
822 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
votes
2answers
242 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 ...
0
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0answers
54 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 ...