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I am having fun with pydrake for the region of attraction (RoA) analysis. There is one type of non-linear system involving trigonometry that got me stuck. A similar problem has been posted here.

As Russ points out in that post, there are 2 solutions: 1) shifting to another state-space; 2) using Taylor expansion. I am applying the 2nd approach to the pendulum case to get a feeling. The system dynamics are given in here. Here is what I have:

1. Linearize the system, and find an LQR, whose cost-to-go function will help to propose a Lyapunov candidate:

I have the following python code for QR factorization. At line of Q[:,i] = u / norm , I get the error mentioned in the title. Can anyone help, please?

• Q is has the shape (3,3),
• u is expected to has the shape (3,1)
• norm is a scalar

The error message is:

I am trying to write this in latex (using overleaf):

I wrote this code:

I want to show a matrix in Rmarkdown pdf file. Using basic LaTeX I created my desired matrix representation:

I want to draw 2 braces under my matrix to indicate that the vectors a are different from the vectors x and explain why. Nevertheless, my current attempt just draws the brace in the middle of my matrice... Does someones has an idea, please? I try to add a snippet here, my code is also here: https://www.overleaf.com/read/nnkpkdrwphmd

Thank you so much for your help!

I'm trying to write some code in R to get from this:

I'm using D3.js to make a double pendulum simulator. I'm using 4th order Runge-Kutta to integrate the ODE's and I'm pretty sure I'm getting the correct values for theta and phi as I checked the output data with mathamatica. However the length of second pendulum keeps changing as it moves which should not be occurring and the path traced out by the pendulum doesn't seem correct either. Since I'm almost certain I am getting the correct values for phi and theta from the integration I think the issue may be how I calculate the x and y coordinates but I'm not sure.

I'm trying to multiply three arrays (A x B x A), with the dimensions (19000, 3) x (19000, 3, 3) x (19000, 3) so that at the end I'm getting a 1d-array with the size (19000), so I want to multiply only along the last one/two dimensions.

I've got it working with np.einsum() but I'm wondering if there is any way of making this faster, as this is the bottleneck of my whole code.

np.einsum('...i,...ij,...j', A, B, A)

I've already tried it with two separated np.einsum() calls, but that gave me the same performance:

np.einsum('...i, ...i', np.einsum('...i,...ij', A, B), A)

As well I've already tried the @ operator and adding some additional axes, but that also didn't make it faster:

(A[:, None]@B@A[...,None]).squeeze()

I've tried to get it working with np.inner(), np.dot(), np.tensordot() and np.vdot(), but these never gave me the same results, so I couldn't compare them.

Any other ideas? Is there any way I could get a better performance?

I've already had a quick look at Numba, but as Numba doesn't support np.einsum() and many other NumPy functions, I would have to rewrite a lot of code.

OK, this question probably has a very simple answer, but I've been searching for quite a while with no luck...

I want to get the dot product of 2 complex numbers in complex-plane-space. However, np.dot and np.vdot both give the wrong result.

Example of what I WANT to do: