Surprises from numerical linear algebra (2010)

http://www.amazon.com/Introduction-Linear-Algebra-Fourth-Gil...

From the book's intro:

>But, suppose we asked a professional mathematician to step back a bit from his habitual way of speaking and write in a more linear fashion? And suppose we even asked more, for example, that he make his writing lively? ... The purpose of this book is to furnish the reader with the first mathemat- ical tools needed to understand one of the pillars of modern mathematics, i.e. linear algebra. The text has been written by a mathematician who has tried to step out of his usual character in order to speak to a larger public. He has also taken up the challenge of trying to make accessible to everyone the first ideas and the first techniques of a body of knowledge that is fundamental to all of science and technology.

As others have suggested, something like Golub and Van Loan is a better choice. The technical notes to go with the LAPACK library, and similar documents from those developing related software, are also likely to be of interest to anyone doing this stuff seriously.

Edit: Let's see if this works: http://news.ycombinator.com/edit?id=2993321.

For those who are just starting on this subject, I highly recommend that you rework the proofs for the bounds between the various matrix norms [1]. These bounds are the building blocks for most interesting analyses in matrix computations. And understanding these analyses are essential if you want to know which algorithm will work best for your problems.

[1]: http://en.wikipedia.org/wiki/Matrix_norm#Examples_of_norm_eq...

But back then, everyone was worried even about well-conditioned systems. As of the mid 1940s the best error bound on cumulative roundoff error was 4^n where n is matrix order. That 4^n would be a multiplier on top of condition number (which at the time had not been formalized, it was Turing who did that).

Anyway, the dependence on matrix size turned out to be order n^2, and in practice more like O(n).

* * *

Also, the bounds on solving

  A x = b

Sometimes for a given A and b, you can compute x above quite accurately, but you can't compute inv(A) accurately and multiply by b to get x. It depends on the how the "difficult directions" in A interact with b. This is one reason numerical analysts always ask, "Do you really need to invert A?"

Because of this, the actual condition number of A (i.e., the norm of A times that of inv(A)) does not enter into the bounds on solving A x = b. The bounds depend on a different norm.

In fact, there are several forms of the error bound -- known as backward and forward errors -- depending on what you need.

* * *

Incidentally, this is the kind of stuff that's covered in Golub and Van Loan, or in Trefethen and Bao, but not in a standard Abstract Linear Algebra book. For instance, Strang's book spends only one chapter on numerical linear algebra.

Why on earth would expect that it would be? I took and enjoyed linear algebra using that text-- it didn't specifically address the QM I deal with as a physicist, but since the author can't predict each student's future, I'd hardly characterise that as a problem with the text!

If it was used as a course on portfolios, ok, bad choice -- but directed against a general linear algebra text, your criticism doesn't seem very valid.

And what you apparently call linear algebra is closer to multivariate statistics...