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).
Questions tagged [integration]
208 questions
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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.
It relies on a simple procedure:
sample the…
Luca
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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 take this limit, without explicitly using the above…
interoception
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4
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4 answers
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, given $N$, the calculation of a bunch of weights…
Vazuoeow
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1 answer
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
$$
t- t_0 = \int ^\psi_{\psi_0} \frac{d\eta}{\eta(1+\kappa\ln \eta + x_0 \eta)}.
$$
Here $t$ represents…
AngusTheMan
- 143
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3
votes
1 answer
Composite simpson's rule with odd intervals
The composite simpson's rule subdivides the interval into n equal subintervals, with n even.
Then
$$\int_a^b f(x) dx \approx \frac{h}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)] $$
(where $x_0 = a$ and…
bernie
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votes
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How to compute line integral over circle
I have a discrete 2D temperature field, i. e. a dataset of N points (x, y, T). I would like to compute line integrals at fixed radii,
$\int_0^{2\pi}T(r,\theta)d\theta$. My first thought was to transform all coordinates into radial coordinates, and…
akid
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votes
1 answer
Numerical integration of sharp peaked function (position of peak known)?
What methods are available to integrate a sharply peaked function (position of peak known) on a finite interval (the interval includes the peak)?
Currently I am getting underflows using some of GSL's adaptive algorithms. I suspect that GSL fails to…
a06e
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vote
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Why can we remove the half-step velocity in velocity Verlet
Eliminating the half-step velocity, this algorithm may be shortend to
Why can we eliminate the half-step velocity and all the math behind the velocity Verlet to what Wikipedia shows?
ght007
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vote
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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 the number of values and their coordinates.
My…
Balfar
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1
vote
1 answer
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)
$$ \int_{\gamma}^{+\infty}f(x) dx = 0.01 $$
The function f isn't a known analytical…
Sam Gomari
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vote
1 answer
Numerical integration of a quadratic form exponential in two variables over a rectangle
Let
$$f(x,y) = \exp \left(- \frac{1}{2}a x^2 - \frac{1}{2}c y^2 + bxy \right)$$
where $a,b,c\ge 0$. I want to integrate numerically:
$$\int_{x_0}^{x_1}\mathrm{d}x \int_{y_0}^{y_1}\mathrm{d}y \, f(x,y) x^ny^m$$
where $-\infty < x_0 < x_1 < \infty$,…
a06e
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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 (spacing distance dz), and r being spaced non-regularly. Moreover, r[0]>0 due to the chosen discretization. For integration, I…
arc_lupus
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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 approximation of the integral.
I'm fairly…
pyhat32