Name: Numerical linear algebra
ISBN: 9780898713619

Numerical linear algebra

An introduction to the field of numerical linear algebra. It aims to present the core, standard material in a novel way. Topics include iterative methods for systems of equations and eigenvalue problems and the underlying principles of conditioning and stability.

What Hacker News says

jacobolusJul 2026

Really? I find the part about the SVD in Axler's book extremely unhelpful, big blobs of opaque formulas and jargon with next to no explanation or context that basically require either knowing the topic fully beforehand or a huge amount of effort to parse.

e.g. Axler's definition of singular values is the extremely dry and technical:

> Suppose T is in L(V, W). The singular values of T are the nonnegative square roots of the eigenvalues of T†T, listed in decreasing order, each included as any times as the dimension of the corresponding eigenspace of T†T.

(Using a dagger instead of an asterisk for the conjugate transpose since HN interprets and asterisk to mean italics.)

If you already just proved a lot of stuff about eigenvalues, this could be a serviceable definition; at any rate it saves space. But it doesn't really explain the point.

I'd recommend anyone interested in this or related topics read Trefethen & Bau (1997) Numerical Linear Algebra.

thread

FilosofumRexApr 2025

The classic reference on the subject is "Numerical Linear Algebra" by Lloyd Trefethen. Skip to the last chapter on the iterative methods for computational aspects. You'll learn a lot more and faster with Matlab.

https://davidtabora.wordpress.com/wp-content/uploads/2015/01...

A short overview is chapter 11 in Gilbert Strangs's Intro to linear Algebra https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_1...

AD comes from a different tradition - dating back to FORTRAN 77 programers attempt to differentiate non-elementary functions (For Loops, procedural functions, Subroutines, etc). Note the hardware specs for some nostalgia https://www.mcs.anl.gov/research/projects/adifor/

thread

blahblahblah10Jan 2023

Out of this list, the books I am familiar with, are great (Hilbert-Courant, Spivak, Korner's books). At the same time, even with extensive mathematical training, I haven't read them from start to finish. I wouldn't even like to say "read". For someone who's not used to mathematical reading, some of these books require careful study. That means generating examples to understand results (theorems), trying your own conjectures, proving things yourself etc. Over time, one becomes familiar with most/all the material in a book but the knowledge might have been acquired through various books (and courses) over time.

Also, mathematics is a massive field. The first question would be what kinds of mathematics would you like to get better at. There are great books in analysis. If you are starting out with a solid calculus knowledge, try Abbott's Understanding Analysis [1] or Duren's Invitation to Classical Analysis [2]. For asymptotic methods in PDEs, try Bender and Orszag [3] which is a wonderful book. But again, this might not be your cup of tea at all and there are more abstract or formal books like Rudin's.

If you want to approach fields without a lot of machinery, graph theory books by Bollobas are great (but difficult). See his Modern Graph Theory book [4] as an example.

For linear algebra, one of my favorites (but it was after I already learned the subject) is Trefethen's Numerical Linear Algebra book [5]. Another beautiful topic is at the intersection of linear algebra and combinatorics. See Babai and Frankl's lectures freely available online.

Then there are wonderful topics in geometry. A massive mountain to climb would be algebraic geometry. For one starting point, see [6]. Differential geometry (Spivak's multi-volume work or Needham's differential forms book) is another wonderful area. I would recommend Crane's discrete differential geometry course at Carnegie Mellon [7] if you want a concrete introduction.

You might want to demystify a topic you have heard about. E.g. Galois theory and the unsolvability of quintic equations. You could look at [8] which guides your way through wonderful problems.

We haven't even touched huge swathes of mathematics including anything topological or number theory. Even within the topics mentioned above, once you start, your journey will take a life of its own and you'll encounter multiple books and papers opening up new sub-fields.

The only approach that worked well for me in the past was to get completely consumed by what one topic one was studying. This meant not getting distracted by multiple topics. Once one enters the workforce, this is very hard (or at least has been for me). Without knowing someone, it's hard to recommend anything but the advantage with topics like graph theory and combinatorics is that one needs less machinery (as opposed to something like algebraic geometry). These fields lead you to interesting problems very rapidly and one can wrestle with them part-time.

[1] https://www.amazon.com/Understanding-Analysis-Undergraduate-...

[2] https://www.amazon.com/Invitation-Classical-Analysis-Applied...

[3] https://www.amazon.com/Advanced-Mathematical-Methods-Scienti...

[4] https://www.amazon.com/Modern-Graph-Theory-Graduate-Mathemat...

[5] https://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefet...

[6] https://www.amazon.com/Algebraic-Geometry-Approach-Mathemati...

[7] https://www.cs.cmu.edu/~kmcrane/Projects/DDG/

[8] https://www.amazon.com/Through-Exercises-Springer-Undergradu...

thread

sfpotterNov 2022

"Numerical Linear Algebra" by Trefethen and Bau.

"Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory" by Bender and Orszag.

I'm not sure if they meet your definition of being well-written exactly as you say above, but these are extremely well-written math books.

This isn't a math book, but I was also a big fan of "Vector Quantization and Signal Compression" by Gersho and Gray.

thread

eigenketOct 2022

You seem to be confusing the mathematical definition of matrix multiplication with an explicit algorithm to compute it. The wiki page you linked is almost entirely about the abstract mathematical definition and talks about algorithms only for a single paragraph, which is about computational complexity. If you look at the wiki page for a particular algorithm (e.g. Strassen: https://en.wikipedia.org/wiki/Strassen_algorithm) then of course they talk about stability in comparison to the naive algorithm (although the page could be improved quite a bit).

If your college textbooks do not mention error analysis, conditioning and stability then they are not numerical linear algebra books worthy of the name. Check out a reference like Trefethen and Bau's Numerical Linear Algebra for example. This book has a whole part (out of the 7 parts in the book) talking about conditioning and stability, and these ideas are present throughout other parts as well.

Once more the type of analysis I'm talking about is emphatically not implementation dependent. It is a property of the algorithm itself. For an example of the sort of statement I mean check out theorem 3.1 of this paper: https://arxiv.org/abs/math/0603207. If you disagree with me then please indicate what sort of "implementation factors" appear in the statement of the theorem.

thread

warinukraineOct 2022

I recommend Numerical Linear Algebra by Trefethen and Bau.

Thanks for this excellent link.

thread