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4

Yes. You can run rank-revealing QR on your matrix $A$, which will stop at step $k$ (hence effectively terminating in $O(mnk)$) and produce $A = QRP$, where $R$ has nonzeros only in its first $k$ rows, and $Q,P$ are orthogonal. You can now compute and SVD of $R$, and use it to piece back the factors with a few matrix products with cost $O(\max(m,n)k^2)$.

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Another solution is to solve with solve_ivp and use the dense_output option which allows interpolating between solution steps: import numpy as np from scipy.integrate import ode, solve_ivp import matplotlib.pyplot as plt import warnings def f(t,y): l = 1 m = 1 d = 1 g = 9.8 return [y, -np.sin(y)*g/l-y*d/m] y0, t0 = [np.pi/2, 0], 0 ...

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Many numerical tips and theoretical explanations can be found in this book from Hairer and Wanner: https://www.springer.com/gp/book/9783540566700 In this book, a strategy is described, which uses a time step such that the relative variation of the solution during the first time step is below a certain threshold if you were using explicit Euler (omitting the ...

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I just found out about Voronoi particle tracking so i'm definitively not an expert. I just want to share what I have found to help others on there journey. The author of the posted Shadertoy examples has a blog where he talks about it: https://michaelmoroz.github.io/Reintegration-Tracking/ Some papers that are talking about it are not behind a paywall (...

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