What are the most widely used C++ vector/matrix math/linear algebra libraries, and their cost and benefit tradeoffs?

So I'm a pretty critical person, and figure if I'm going to invest in a library, I'd better know what I'm getting myself into. I figure it's better to go heavy on the criticism and light on the flattery when scrutinizing; what's wrong with it has many more implications for the future than what's right. So I'm going to go overboard here a little bit to provide the kind of answer that would have helped me and I hope will help others who may journey down this path. Keep in mind that this is based on what little reviewing/testing I've done with these libs. Oh and I stole some of the positive description from Reed.

I'll mention up top that I went with GMTL despite it's idiosyncrasies because the Eigen2 unsafeness was too big of a downside. But I've recently learned that the next release of Eigen2 will contain defines that will shut off the alignment code, and make it safe. So I may switch over.

Update: I've switched to Eigen3. Despite it's idiosyncrasies, its scope and elegance are too hard to ignore, and the optimizations which make it unsafe can be turned off with a define.

Eigen2/Eigen3

Benefits: LGPL MPL2, Clean, well designed API, fairly easy to use. Seems to be well maintained with a vibrant community. Low memory overhead. High performance. Made for general linear algebra, but good geometric functionality available as well. All header lib, no linking required.

Idiocyncracies/downsides: (Some/all of these can be avoided by some defines that are available in the current development branch Eigen3)

• Unsafe performance optimizations result in needing careful following of rules. Failure to follow rules causes crashes.
  • you simply cannot safely pass-by-value
  • use of Eigen types as members requires special allocator customization (or you crash)
  • use with stl container types and possibly other templates requir

I've heard good things about Eigen and NT2, but haven't personally used either. There's also Boost.UBLAS, which I believe is getting a bit long in the tooth. The developers of NT2 are building the next version with the intention of getting it into Boost, so that might count for somthing.

My lin. alg. needs don't exteed beyond the 4x4 matrix case, so I can't comment on advanced functionality; I'm just pointing out some options.

Okay, I think I know what you're looking for. It appears that GGT is a pretty good solution, as Reed Copsey suggested.

Personally, we rolled our own little library, because we deal with rational points a lot - lots of rational NURBS and Beziers.

It turns out that most 3D graphics libraries do computations with projective points that have no basis in projective math, because that's what gets you the answer you want. We ended up using Grassmann points, which have a solid theoretical underpinning and decreased the number of point types. Grassmann points are basically the same computations people are using now, with the benefit of a robust theory. Most importantly, it makes things clearer in our minds, so we have fewer bugs. Ron Goldman wrote a paper on Grassmann points in computer graphics called "On the Algebraic and Geometric Foundations of Computer Graphics".

Not directly related to your question, but an interesting read.