# Tag Info

Accepted

### What does "symplectic" mean in reference to numerical integrators, and does SciPy's odeint use them?

Let me start off with corrections. No, odeint doesn't have any symplectic integrators. No, symplectic integration doesn't mean conservation of energy. What does ...
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### Which Runge-Kutta method is more accurate: Dormand-Prince or Cash-Karp?

Since I just finished optimizing a lot of them in software, DifferentialEquations.jl, I decided to just lay out a comparison of the main Order 4/5 methods. The Fehlberg method was left out because it'...

### What does "symplectic" mean in reference to numerical integrators, and does SciPy's odeint use them?

To complement Chris Rackauckas answer, to state some of the mathematical nonsense as well as some stuff you almost certainly know, a dynamical system is Hamiltonian if there is a description with ...
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### CUDA & Python for numerical integration and solving differential equations

Julia's DifferentialEquations.jl is all GPU-compatible. If you make your arrays GPU-based arrays, then the solver recompiles to be all on the GPU (no data transfers). For example: ...

### Energy conservation in RK4 integration scheme in C++

RK4 is not symplectic so it has no guarantee of energy conservation. Especially when solving an N-body problem where two bodies pass by close to each other the energy conservation can be violated ...
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### How can I numerically integrate the Kepler problem?

I'm going to assume for the moment that your code is correctly implemented and that this problem isn't a bug. Believe it or not, gradual increase of the energy is the expected behavior of most simple ...

### Numerical evaluation of highly oscillatory integral

Use Plancherel's theorem to evaluate this integral. The basic idea is that for two functions $f,g$, $$I=\int_{-\infty}^{\infty} f(x) g^*(x)dx = \int_{-\infty}^{\infty} F(k) G^*(k) dk$$ where $F,G$...

### Integrating Lagrange polynomials with many nodes, round-off

You can evaluate this using the Björck-Pereyra algorithm for solving Vandermonde systems, because you are evaluating $b^\top V^{-1}$ with $b=(2,0,\frac23,0,\frac25,0,\ldots)$, and the algorithm is ...
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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

That is a common misconception. Verlet in whatever form is a second-order method. The misconception results from the fact that the truncation error of the Verlet formula is of 4th order. The naive ...
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### How to solve the problem without using symbolic computation

You can solve this numerically in Python without symbolic computation. from __future__ import print_function, division import numpy as np from numpy import exp from scipy.integrate import quad from ...
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### Numerical integration of a hypergeometric function

First of all, from the first paragraph of your attempts at a solution, I assume that the $z_j$ are non-negative? In that case, the integrand has no real problematic points (it's monotonous, decreasing)...
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### What is wrong with this Euler method code in python?

I did not check your code, however, the result you are getting is also verified by scipy.integrate.odeint ...
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### How to estimate the error of trapezoidal rule using discrete data?

You may use the ideas of error extrapolation as one uses it to construct high-order Runge Kutta methods. Depending on the function that you interpolate, the interpolation error $I - I_h$, where $I$ ...
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### Evaluating the surface integral in an FEM (Finite Elements Method) procedure

The particular surface integral you want to calculate is basically a specific case of integrating a function over a surface defined in terms of two parametric coordinates. Lets first consider this ...
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### Integration of the Fermi distribution using Python

One of your problems is the system of units that you are using. Just changing the units improves the results ...
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### How do I integrate a function defined over an arbitrary area?

Instead of directly integrating over the area, it is often more convenient to use the divergence theorem to replace the area integral with an integral over the boundary edges. The divergence theorem ...

### Numerical evaluation of highly oscillatory integral

The key to the evaluation of oscillatory integrals is to truncate integral at the right point. For this example you need to choose upper limit of the form $$\pi\mathbb{N}+\frac{\pi}{2}$$ Before ...

### Trapezoid rule vs Gaussian quadrature: what am I missing?

How accurate do you want the answer ? How costly is evaluating your function ? If it is costly, then you dont want to use a rule with too many nodes. How many times do you want to do the quadrature ?...

### Numerical integration giving incorrect sign

This might be an accuracy problem in computing the second term, because of those large exponentials when $x \gg 1$. I would first work on that term: gather $e^x$ out from numerator and denominator and ...

### Energy conservation in RK4 integration scheme in C++

Main error As I pointed out in the previous questions of this series RK4 integration of the three-bodies problem with C++ the primary problem is that the methods are not implemented correctly. The ...
Accepted

### Looking for an accurate algorithm to find the area of a oscillatory function

You can look up various quadratures. One method that should fair better is Gauss Quadrature. I would also recommend looking into any adaptive quadrature schemes. There are many of them out there, so ...

### Integral in log-log space

You can just change variables. Setting $a=log(x)$, $b(a)=log(y(x))$. The integral becomes $F(r)=\int^{log(r)}_{-\infty} exp(a+b) da$ You have to be a little careful because you are integrating from ...