Questions tagged [integration]

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49 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}^{...
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0answers
8 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
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1answer
58 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
73 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 ...
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30 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
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1answer
96 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 ...
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1answer
124 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^{\...
3
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1answer
40 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 ...
3
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1answer
73 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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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 ...
3
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1answer
162 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 ...
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0answers
69 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 ...
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2answers
191 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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0answers
94 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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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}...
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1answer
75 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: ...
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1answer
96 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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0answers
35 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 ...
3
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0answers
45 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 ...
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0answers
20 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 ...
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0answers
74 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)/...
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0answers
106 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}\...
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3answers
255 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 ...
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1answer
146 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. ...
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3answers
83 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
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1answer
378 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 ...
2
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1answer
88 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'...
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1answer
72 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
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1answer
78 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
51 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}_{\...
3
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1answer
91 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+...
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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
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4answers
571 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
215 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 ...
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0answers
73 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
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 ...
3
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2answers
213 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
71 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
84 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
52 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
85 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 ...
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2answers
54 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 ...
3
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0answers
40 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
137 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
250 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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3answers
378 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
89 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, ...
1
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1answer
101 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 ...