Just a quick update on the problems the authors encountered in the Durán et al. article:

As we all know, there's an infinite amount of math computations out there, and we do a tremendous amount of testing but it's always possible to miss something. We thank the article's authors for pointing out these particular bugs which have now been fixed with last week's Mathematica 10.0.2 release.

I should note that we ran a blog post in 2007 by Stephen Wolfram that addresses these kinds of issues, and is worth revisiting:

Mathematics, Mathematica, and Certainty

A relevant extract:

"We run millions and millions of tests on every version of Mathematica, trying to exercise every part of the system. And doing that is orders of magnitude more powerful at catching bugs than any kind of pure human testing.

Sometimes we use the symbolic capabilities of Mathematica to analyse the raw code of Mathematica. But pretty quickly we tend to run right up against undecidability: there’s a theoretical limit to what we can automatically verify."

If folks here are interested in more details surrounding the particular issues in question, I could try and see about getting one of our senior developers to write something up (no promises, as we are all pretty busy).

My Windows 8.1 64 bit computer running Mathematica 10.0.0.0 gives the same value 26727055876061626.... for Det[bigMatrix] obtained previously by Udo Krause and David Park Jr. I have also written a program that diagonalizes an integer matrix by row and column reductions. If the matrix is square the product of the diagonal entries is then the determinant. This gives the same answer, so there is little doubt we have the correct value. Mathematica 9.0.0.0 running on the same computer alternates at random between two values for Det[bigMatrix], both incorrect, and one exactly twice the other. These appear to be the values 2.284217140 x 10^9769 and 1.142108570 x 10^9769 mentioned by David Park Jr.

Hi David, I agree with the general approach and attitude outlined in the first paragraph of your post. But as Frank Kampas indicated, there is a certain "triage" perspective that has to be taken when a product like Mathematica is developed.

More precisely, we can say that the "hierarchical approach" you talk about is basically transparency of the algorithms. That transparency of the "high-power routines" can be done in two ways: (i) with API's of sub-routines, as you suggested, and (ii) by detailed documentation with computation examples, comparisons, etc. WRI tries to do both, but WRI's "triage" might produce different outcomes than what ambitious or power users might expect.