# Questions tagged [numerical-integration]

Questions on the use of numerical functions NIntegrate and NDSolve.

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### When can I assume that all decimal digits returned by Mathematica are provably correct?

How to Control the Precision and Accuracy of Numerical Results Arbitrary-Precision Numbers Mathematica works with exact numbers and with two different types of approximate numbers: machine-...
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### Nested NIntegrate

Suppose that we have the given simple integral expression $$\int_{-5}^{5} x \int_{-\infty}^{x} e^{\int_{0}^{z} -y dy} dz dx$$ Writing this out in Mathematica we obtain: ...
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### Complex valued 2+1D PDE Schrödinger equation, numerical method for NDSolve?

Based on the heat equation of the Mathematica Manual tutorial, I wrote the complex counterpart (Schrödinger) equation, for the free particle propagation of an initial wavepacket. ...
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### How to implement custom integration rules for use by NIntegrate?

How can NIntegrate be extended with custom implementation of integration rules? This answer of the question "Monte Carlo integration with random numbers generated ...
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### Determining which rule NIntegrate selects automatically

I need to numerically integrate a highly oscillatory function over the semi-infinite domain $(0,\infty)$: $$\int_0^\infty \frac{\sin^2(x) \sin^2(1000 x)}{x^{5/2}}\mathrm dx$$ Since the Levin rule (...
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### Numerical integration --- Mathematica vs Python (w/ Scipy) performance

I'm about to tackle a problem that involves a lot of (multi-dimensional) numerical integrations and also subsequent optimizations, and so I want to make sure at least the integration step is as fast ...
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### How to use NDSolve to track equilibrium?

I am looking for an extension of NDSolve where integration runs until certain variables are settled at an equilibrium. Now I have a working solution in my sleeves ...
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### Plotting separatrices for nonlinear system

Consider the system: \begin{align*} x'&=(1-x-y)x\\ y'&=(4-7x-3y)y \end{align*} The system has a saddle point at (1/4,3/4). How can I plot the separatrices on the phase portrait having domain ...
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### Numerical inverse Laplace-Hankel transform

When trying to reproduce the result of this paper about numerical solution of Lamb's problem, I encountered the following double integral (to be more precise, the 0-order inverse Hankel-Laplace ...
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### How to speed up the plot of NIntegrate?

Here is a toy example: f[t_] := NIntegrate[Sin[x], {x, 0, t}]; Plot[f[t], {t, 0, 10}] // Timing Even such a simple example will take 2.8 seconds on my computer. ...
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### Only final result from NDSolve

Finally, I started to play with differential equations in Mathematica. And I have faced the problem, which seems to me so basic that I'm afraid this question is going to be closed soon. However, I'...
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### How to calculate the volume of a convex hull?

Given a spatial curve represented by a parametric equation, is it possible in Mathematica 9 to calculate symbolically (or at least numerically) the volume of its convex hull?
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### NDSolve uses different difference order for different spatial derivative when solving PDE

I found something this tutorial for method of line doesn't tell us. Consider the following toy example: ...
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### More efficient method to compute moments of the Johnson $S_B$ distribution

Here is a very specific feature request. I need Mean[JohnsonDistribution["SB", γ, δ, 0, 1]] When I issue e.g. ...
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### What's behind Method -> {"EquationSimplification" -> "Residual"}

In order to solve a quite large system of differential equations, I have the habit to use the NDSolve command without changing any options. As I wanted more ...
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