# 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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### Am I doing leapfrog integration correctly?

I wrote this minimal example to examine the Leapfrog integration algorithm. However, I am not sure it is the correct algorithm, and the listing is giving the correct output. Is this the Leapfrog ...
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### Understanding leapfrog integration algorithm

The leapfrog.cpp is an implementation of leapfrog integration algorithm where f() function is being integrated: leapfrog.cpp <...
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### Solving system of ODEs, where time derivative approaches infinity due top initial condition

I am trying to solve a problem in python using scipy's solve_ivp. The system of ODEs I am trying to solve is for coupled where I am solving for two time-dependent ...
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### 2D integrals in Python with specified points of interest

Note: This is my first question on stackexchange; please tell me if I'm doing something incorrectly. I am trying to calculate a series of a 2D integrals in Python with an integrand that has several ...
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### Solving IVP backward in time via python

I'm having difficulty solving an initial value problem (IVP) in Python backwards in time. The code is at the end of this post. First, please let me state my simplified problem. The forward IVP is ...
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### Matching the limits of integration with the proper variables in a complicated case when using scipy.integrate.nquad

I need to integrate expressions containing powers of the function: ...
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### 'Integral2' error in MATLAB for invalid integrand

Here is the code that I am trying to run: ...
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### 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 ...
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### Integration of a singular kernel function over a triangle

Problem: I am currently trying to integrate a singular kernel function of the type $$G(x,y)=\frac{\exp(ik||x-y||_2)}{4\pi ||x-y||_2}$$ which lies at the centre of a triangle, over this triangle. $i$ ...
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### B-spline abscissa values for numerical stability

I am working with spline functions using the B-spline basis, $$f(x) = \sum_{i=1}^n c_i B_i(x;\mathbf{t})$$ where the $B_i$ are cubic B-splines with a knot vector $\mathbf{t}$. For my application, ...
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### 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 ...
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### 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 ...
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### Using the Kramers-Kronig (Hilbert) transform in Python

I am trying to use the Kramers-Kronig algorithm to transform the real and imaginary contributions to the anomalous scattering factor from a diffraction anomalous fine structure (DAFS) experiment. I ...
1 vote
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### RK4 integration of the three-bodies problem with C++

first of all thank you for all the answers you gave me yesterday for the integration via Symplectic Euler's method of the three-body problem. We managed to implement both Euler's and Runge Kutta 4's ...
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### Quadrature rules for products of 2D regions

I am interested in computing integrals of the form $\iint_{P\times P} Q(x_1,x_2,y_2,y_2) dxdy$ where $P$ is a polygon and $Q$ is a polynomial. The coordinates $(x_1,x_2)$ are in the plane of $P$. Of ...
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### Numerical integration of a 2D hemisphere discrete dataset where limits are unknown

I am trying to compute the integral of a 2D hemisphere dataset $f_r \, (\theta, \phi)$ where $\theta \in [0, \pi / 2[$ and $\phi \in [0, 2\pi]$. I am making the measurements myself, so I can choose ...
1 vote
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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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### 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 ...
1 vote
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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 ...
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 ...
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 ...