[dagss]

For programmers, the really interesting part of dense linear algebra is how to achieve high performance, as blocking techniques have to be used to amortize loads from memory to cache.

Google for Goto's "Anatomy of high-performance matrix multiplication", one of my favorite programming texts.

Also the papers underlying the development of the "Elemental" library for distributed dense linear algebra is worth a look.

[gmiller123456]

To save people a few clicks:

[1] https://www.cs.utexas.edu/users/pingali/CS378/2008sp/papers/...

[FabHK]

BTW, I think one important take-away for programmers is:

Don't invert matrices.

For example, the solution for the least squares (=linear regression) problem is

  beta^hat = (X'X)^-1 X'y

and you might think that the best way to compute it is to compute X'X, invert it, do the multiplication, yada yada yada. But it's really better to just ask your library (Matlab, LAPACK, ...) to solve the problem for you, and it'll probably do a QR decomposition. For small problems, it doesn't really matter, but for big problems you'll be both faster and more accurate.

[R_haterade]

Excellent John D. Cook piece on the same subject: https://www.johndcook.com/blog/2010/01/19/dont-invert-that-m...

[conistonwater]

That's a slightly different issue: the issue with solving normal equations directly is that the condition number of X^\top X is the square of the condition number of X, so it's much more significant than lu-vs-inv. It can matter a great deal for ill-conditioned linear regression problems where some features resemble each other.

[FabHK]

> how to achieve high performance

You really just want to use LAPACK/BLAS, no? (That's what the Neanderthal library mentioned in the article does, btw, and basically linear algebra libraries for other languages, too. If not, you probably shouldn't use it...)

http://neanderthal.uncomplicate.org

[dagss]

Of course you should but the blocking and amortization techniques have much wider usecases beyond linear algebra. Matrix multiplicaton is a useful "simple" example to study; of course you don't write your own if what you want to do is covered by BLAS/LAPACK.

For instance I have written high performance spherical harmonic transforms, and these techniques apply then.