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Accepted

Why do we usually not want the eigenvalues of non-symmetric matrices?

Stability under perturbations Let $E$ be a perturbation such that $\|E\| \leq \varepsilon$. If $A$ is symmetric, then the eigenvalues of $A+E$ are at a distance $\varepsilon$ from those of $A$. (Bauer-...
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Is half precision supported by modern architecture?

Intel support for IEEE float16 storage format Intel supports IEEE half as a storage type in processors since Ivy Bridge (2013). Storage type means you can get a memory/cache capacity/bandwidth ...
• 2,096
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Conserving Energy in Physics Simulation with imperfect Numerical Solver

There are a few ways to conserve energy during ODE integration. Method 1: Symplectic Integration The cheapest way that is to use a symplectic integrator. A symplectic integrator solves the ODE on a ...
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Scientific computing vs numerical analysis

Wikipedia gives a good definition Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical ...
• 18.2k
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Use of machine learning in computational fluid dynamics

It's a long-running joke that CFD stands for "colorful fluid dynamics". Nevertheless, it is used -- and useful -- in a wide range of applications. I believe your discontent stems from not ...
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Why is RK45 used as the "default" method for non-stiff ODEs rather than a multistep one?

First, let's establish that they are a good choice. The SciMLBenchmarks are probably the most comprehensive that there are as of right now for modern methods. This uses the vast number of methods ...
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Why aren't Krylov subspace methods popular in the Machine Learning community compared to Gradient Descent?

On a basic level, I don't buy the argument that you have to "solve a linear system for many machine learning algorithms". Much more, you usually have to optimize a non-linear equation which ...
• 2,937
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Compute powers close to zero

Use the Taylor series expansion of $a^x$ about $x=0$ and evaluate a small number of terms for $a=10$: $$a^x - 1 \approx x\log(a) + \frac{1}{2}x^2\log^2(a)+\cdots.$$

Compute powers close to zero

Personally, in the absence of a special function like expm1 (see also this scicomp.SE question), I'd use a Padé approximant instead of a Taylor/Maclaurin series; ...
• 3,115
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Why is the central difference method dispersing my solution?

I'll write the equation short as $$\ddot x(t)+c\dot x(t)=a(t,x(t))$$ to separate the "easy" linear parts from the non-linear and forcing terms. On the first method The claimed order of the ...
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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 ...
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Runge-Kutta in the presence of an attractor

The problem you are encountering is likely not a consequence of your choice of algorithm, but in fact a consequence of the resulting dynamical system after applying time reversal. Per the definition ...
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How important is learning hardware/architecture for scientific computing?

I haven't worked in quantum chemistry specifically, but I've worked in other areas where high performance is a correctness requirement (along with scientific accuracy), so I think we're on the same ...
• 351
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Analytical convergent sequence and numerical divergent sequence

Jean-Michel Muller, et. al., "Handbook of Floating-Point Arithmetic 2nd ed.", Birkhäuser 2018, gives the following example due to Muller, specifically constructed to deliver incorrect results with ...
• 1,320
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Going back in time in an initial value problem

This is technically still an IVP if you do an appropriate change of variables. Given your time is between $t \in [t^*, 0]$, make a new time variable $\tau = -t$ so that $\tau \in [0, -t^*]$ and you ...
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What are the most important theorems in computational science?

You'll get everyone to give different answers to this question, and maybe that's alright. Here are some of my favorite ones: Taylor's theorem that a function (of sufficient smoothness) equals its ...
• 52.4k

How to solve a second order differential equation (diffusion) with boundary conditions using Python

I have found that I must keep the value of dt near dx or the results become unstable This behavior you have noticed is known as the Courant–Friedrichs–Lewy (CFL) condition. Indeed, there are ...