docs: Add triangular solve figure

[thomaspinder]

In our sharp bits doc, we acknowledge that solving a system of equations using the Cholesky factor is faster than using the dense covariance matrix. It would be nice to have a figure that depicts why this is true.

[ya-sh-vardh-an]

Hi, this is my first issue but I am really interested in it. How can I apply?

[thomaspinder]

Hi @Yash5675, thanks for the interest. For this section in our sharp bits doc, it would be nice to have an example of solving a small (3x3) system of equations using Cholesky factors.

If I have a PSD matrix $A=LL^{\top}$ and I want to solve $Ax=y$, then I can first solve $Lx=b$ and then $L^{\top}x=b$. These two operations both involve triangular matrices and are, therefore, faster than brute force solving $Ax=y$. If this is new to you, then there are some nice details on this in Chapter 26 of Numerical linear algebra.

It would be nice to have something that elucidates this idea. Ideally, this would be a figure, but also a toy math example would be fine too. You can include this into the sharp bits doc and open a PR. If you have any questions, then please don't hesitate to message. We do also have a Slack channel.

[ya-sh-vardh-an]

@thomaspinder thank you for the great explanation and materials for more information. I have some questions. Shall I ask them here or on slack? I am fine with either.

[thomaspinder]

Wherever you prefer. If your questions are package-specific, then posting here may be better for transparency. Otherwise, Slack is fine :)

[ya-sh-vardh-an]

Ok, I will just ask them here. So, what I have to do is add an illustration into docs/sharp_bits.md at the end of the Positive-definiteness section. Please correct me if I am wrong. About the illustration, it should show Cholesky factorisation steps:

1. Cholesky Decomposition
2. Forward Substitution
3. Backward Substitution

through an example of 3x3 system of equations? (should I also include the operation count?). And ideally it should be a figure (should I add the figure into the docs/_static/ and use the link from there?).

PS: I have no idea what I am saying. Please correct me if any of the information is wrong or needs more improvement.

[thomaspinder]

Correct. If you are able to include the operation count, then that would be excellent! A figure would be ideal and yes, it can go under docs/_static/ for now.

Feel free to exercise your creativity here - if, whilst doing this, you feel there's a better way to illustrate the point, then you should free to do this. We can manage any imperfections or areas for improvement through the PR.

[ya-sh-vardh-an]

Sorry for the late reply. I have opened a PR. Please let me know what changes I need to make. Thank you very much.

[ya-sh-vardh-an]

Hello @thomaspinder, I have made an animated version of the example. (the preview is posted in the PR #276 ) Should I open a new PR or reopen the previous one.

PS. I am really sorry for the late reply !

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