Recently released book using Julia

[kylenovak29]

Kyle A. Novak, Numerical Methods for Scientific Computing: The Definitive Manual for Math Geeks, 2nd Edition. Equal Share Press, March 2022. The book is also available for free as a PDF from equalsharepress.com.

[ViralBShah]

Would be great if you could edit this page: https://github.com/JuliaLang/www.julialang.org/blob/main/learning/books.md

[ViralBShah]

@kylenovak29 You have a really interesting background. Thanks for putting this book together. I enjoyed flipping through it!

I did notice that you are using LightGraphs for some of your examples, and that package has been updated to Graphs. Not sure if your book is a live book and can be updated - or if it may be something to be addressed in a second edition.

[kylenovak29]

Added. Thanks.

[kylenovak29]

Thanks for the feedback! Yes, I can update it. I'm using an agile publishing methodology to make the book available with minimal errors and improve it with feedback. The book is live and it can be updated (both the PDF and the print edition). I should be able to make the changes to the book to use Graphs instead of LightGraphs within the next few days. If you happen across any other errors/suggestions at any time, let me know. Contact information is in the edition page of the book.

[ViralBShah]

@ChrisRackauckas @YingboMa - @kylenovak29 's book also has a nice chapter on ODEs with a nice comparison table. Since we have a line with the author - I thought I would ping you if you wanted to take a quick look and make suggestions, if any.

[kylenovak29]

Many thanks to @ChrisRackauckas, who has already provided some helpful comments on correcting/improving the comparison table (which has since been updated). Any additional feedback is welcome.

[YingboMa]

Thanks a lot for writing the book! I really enjoyed reading it.

BTW, IDA, ode15i, and BDF are all BDF not NDF. We also have the the pure Julia implementation DFBDF which is mathematically equivalent with ode15i. QNDF is the equivalent of ode15s, and both of them are NDF.

Also, Rosenbrock23 and ode23s are both order 2.

[ChrisRackauckas]

I would simplify that down to:

BDF/NDF Julia: CVODE/FBDF/QNDF MATLAB: ode15s Python: BDF

BDF (DAE) Julia: IDA/DFBDF MATLAB: ode15i Python: ------

[ChrisRackauckas]

And for reference https://www.reddit.com/r/Julia/comments/tenqrz/book_numerical_methods_for_scientific_computing/i0qt7nv/?utm_source=reddit&utm_medium=web2x&context=3

[YingboMa]

There's no NDF solver for fully implicit DAEs in MATLAB, Julia, or Python.

[ChrisRackauckas]

Yeah that was just a typo

[ChrisRackauckas]

In fact, even mentioning NDF is probably not a good idea for this book. NDF is such a weird obscure thing. Maybe just

BDF Julia: CVODE/FBDF/QNDF MATLAB: ode15s Python: BDF

BDF (DAE) Julia: IDA/DFBDF MATLAB: ode15i Python: ------

Is good enough.

[kylenovak29]

Thanks! I'll add Chris' latest suggestion to the table---it looks like it will all still fit. My hope for the table is primarily a summary of the methods discussed earlier in the chapter and secondly as a quick comparison reference. I don't discuss NDFs or Klopfenstein--Shampine except in passing reference to the table, included for completeness as a hold-over from an earlier edition that focused on Matlab. It's too nuanced and dropping mention of it improves readability. Differentiating BDF with DAE will also fit in well when I add the section on DAEs. Also, thanks for catching the order of Rosenbrock23.

[kylenovak29]

I've updated the table and the section on ODE solvers. To keep the table as simple as possible, I decided to list only pure Julia routines under the Julia column because the other routines in the table could already be used in Julia using a wrapper. I also decided to change the last column to efficiency instead of accuracy because accuracy is arbitrary using adaptive step sizes. I used the benchmarking tools in DiffEqDevTools.jl on two standard stiff/non-stiff problems to roughly categorize the routines by efficiency. Of course, the choice of problems is still arbitrary. I think the table now provides a fairly accurate summary of the methods common across Julia, Python, and Matlab.

[ChrisRackauckas]

VCABM is a Adams-Bashforth-Moulton, same as LSODE and CVODE with Adams coefficients, which is the same as ode113. So those rows should be merged. CVODE also does BDF coefficients (as does LSODE) for stiff equations: they just swap which coefficients they use (and linear solver).

[kylenovak29]

I thought that ode113 was an ABM PECE method (using PECE functional iteration) while LSODE and CVODE were AM methods (with Newton or functional iteration) [or BDF methods (depending on the flag)]. I separated them because the LSODE and CVODE can take/compute Jacobians while ode113 does not. The documentation on VCABM says it uses DDEABM (which doesn't appear to use any type of Newton iteration), so I grouped it with ode113. Should I instead group VCABM with LSODE/CVODE? Or do you think all the methods are similar enough that nuance isn't warranted in the table?

[ChrisRackauckas]

Ehh I wouldn't make a difference there. LSODE and CVODE are ABM methods. They solve a nonlinear system and use Adams-Bashforth for the predictor. Then they default to using functional iteration, which if it only did one step of it you'd call that ABM PECE, while it can now iterate that to convergence (or you can set the max to one). You can only give it a Jacobian and force it to do Newton, but effectively no one uses the Adams method like that because it increases the cost but still isn't good for stiff equations (the BDF coefficients are necessary for that), so it's almost used as an ABM PECE method. So in the end, while a few are a bit more flexible, the flexibility is somewhat meaningless and I would just clump the four together as effectively the same non-stiff ODE solver. There are more differences in the stiff ODE case, but the non-stiff algorithms are almost the same.

[kylenovak29]

Okay. Thanks. I grouped the ABM methods together as well as the BDF methods.