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

There are quite a few projects that have settled on the Generic Graphics Toolkit for this. The GMTL in there is nice - it's quite small, very functional, and been used widely enough to be very reliable. OpenSG, VRJuggler, and other projects have all switched to using this instead of their own hand-rolled vertor/matrix math.

I've found it quite nice - it does everything via templates, so it's very flexible, and very fast.

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Edit:

After the comments discussion, and edits, I thought I'd throw out some more information about the benefits and downsides to specific implementations, and why you might choose one over the other, given your situation.

GMTL -

Benefits: Simple API, specifically designed for graphics engines. Includes many primitive types geared towards rendering (such as planes, AABB, quatenrions with multiple interpolation, etc) that aren't in any other packages. Very low memory overhead, quite fast, easy to use.

Downsides: API is very focused specifically on rendering and graphics. Doesn't include general purpose (NxM) matrices, matrix decomposition and solving, etc, since these are outside the realm of traditional graphics/geometry applications.

Eigen -

Benefits: Clean API, fairly easy to use. Includes a Geometry module with quaternions and geometric transforms. Low memory overhead. Full, highly performant solving of large NxN matrices and other general purpose mathematical routines.

Downsides: May be a bit larger scope than you are wanting (?). Fewer geometric/rendering specific routines when compared to GMTL (ie: Euler angle definitions, etc).

IMSL -

Benefits: Very complete numeric library. Very, very fast (supposedly the fastest solver). By far the largest, most complete mathematical API. Commercially supported, mature, and stable.

Downsides: Cost - not inexpensive. Very few geometric/rendering specific methods, so you'll need to roll your own on top of their linear algebra classes.

NT2 -

Benefits: Provides syntax that is more familiar if you're used to MATLAB. Provides full decomposition and solving for large matrices, etc.

Downsides: Mathematical, not rendering focused. Probably not as performant as Eigen.

LAPACK -

Benefits: Very stable, proven algorithms. Been around for a long time. Complete matrix solving, etc. Many options for obscure mathematics.

Downsides: Not as highly performant in some cases. Ported from Fortran, with odd API for usage.

Personally, for me, it comes down to a single question - how are you planning to use this. If you're focus is just on rendering and graphics, I like Generic Graphics Toolkit, since it performs well, and supports many useful rendering operations out of the box without having to implement your own. If you need general purpose matrix solving (ie: SVD or LU decomposition of large matrices), I'd go with Eigen, since it handles that, provides some geometric operations, and is very performant with large matrix solutions. You may need to write more of your own graphics/geometric operations (on top of their matrices/vectors), but that's not horrible.

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 required special allocation customization (or you will crash)
  • certain compilers need special care to prevent crashes on function calls (GCC windows)

GMTL

Benefits: LGPL, Fairly Simple API, specifically designed for graphics engines. Includes many primitive types geared towards rendering (such as planes, AABB, quatenrions with multiple interpolation, etc) that aren't in any other packages. Very low memory overhead, quite fast, easy to use. All header based, no linking necessary.

Idiocyncracies/downsides:

• API is quirky
  • what might be myVec.x() in another lib is only available via myVec[0] (Readability problem)
    • an array or stl::vector of points may cause you to do something like pointsList[0][0] to access the x component of the first point
  • in a naive attempt at optimization, removed cross(vec,vec) and replaced with makeCross(vec,vec,vec) when compiler eliminates unnecessary temps anyway
  • normal math operations don't return normal types unless you shut off some optimization features e.g.: vec1 - vec2 does not return a normal vector so length( vecA - vecB ) fails even though vecC = vecA - vecB works. You must wrap like: length( Vec( vecA - vecB ) )
  • operations on vectors are provided by external functions rather than members. This may require you to use the scope resolution everywhere since common symbol names may collide
  • you have to do
      length( makeCross( vecA, vecB ) )
    or
      gmtl::length( gmtl::makeCross( vecA, vecB ) )
    where otherwise you might try
      vecA.cross( vecB ).length()
    

• not well maintained
  • still claimed as "beta"
  • documentation missing basic info like which headers are needed to use normal functionalty
    • Vec.h does not contain operations for Vectors, VecOps.h contains some, others are in Generate.h for example. cross(vec&,vec&,vec&) in VecOps.h, [make]cross(vec&,vec&) in Generate.h

• immature/unstable API; still changing.
  • For example "cross" has moved from "VecOps.h" to "Generate.h", and then the name was changed to "makeCross". Documentation examples fail because still refer to old versions of functions that no-longer exist.

NT2

Can't tell because they seem to be more interested in the fractal image header of their web page than the content. Looks more like an academic project than a serious software project.

Latest release over 2 years ago.

Apparently no documentation in English though supposedly there is something in French somewhere.

Cant find a trace of a community around the project.

LAPACK & BLAS

Benefits: Old and mature.

Downsides:

• old as dinosaurs with really crappy APIs

For what it's worth, I've tried both Eigen and Armadillo. Below is a brief evaluation.

Eigen Advantages: 1. Completely self-contained -- no dependence on external BLAS or LAPACK. 2. Documentation decent. 3. Purportedly fast, although I haven't put it to the test.

Disadvantage: The QR algorithm returns just a single matrix, with the R matrix embedded in the upper triangle. No idea where the rest of the matrix comes from, and no Q matrix can be accessed.

Armadillo Advantages: 1. Wide range of decompositions and other functions (including QR). 2. Reasonably fast (uses expression templates), but again, I haven't really pushed it to high dimensions.

Disadvantages: 1. Depends on external BLAS and/or LAPACK for matrix decompositions. 2. Documentation is lacking IMHO (including the specifics wrt LAPACK, other than changing a #define statement).

Would be nice if an open source library were available that is self-contained and straightforward to use. I have run into this same issue for 10 years, and it gets frustrating. At one point, I used GSL for C and wrote C++ wrappers around it, but with modern C++ -- especially using the advantages of expression templates -- we shouldn't have to mess with C in the 21st century. Just my tuppencehapenny.