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

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It seems that many projects slowly come upon a need to do matrix math  and fall into the trap of first building some vector classes and slowly adding in functionality until they get caught building a half-assed custom linear algebra library  and depending on it    I d like to avoid that while not building in a dependence on some tangentially related library  e g  OpenCV  OpenSceneGraph     What are the commonly used matrix math linear algebra libraries out there  and why would decide to use one over another  Are there any that would be advised against using for some reason  I am specifically using this in a geometric time context  2 3 4 Dim   but may be using higher dimensional data in the future    I m looking for differences with respect to any of  API  speed  memory use  breadth completeness  narrowness specificness  extensibility  and or maturity stability   Update  I ended up using Eigen3 which I am extremely happy with

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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        nbsp  nbsp length  makeCross  vecA  vecB           or       nbsp  nbsp gmtl  length  gmtl  makeCross  vecA  vecB        where otherwise you might try        nbsp  nbsp 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

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I m new to this topic  so I can t say a whole lot  but BLAS is pretty much the standard in scientific computing  BLAS is actually an API standard  which has many implementations  I m honestly not sure which implementations are most popular or why   If you want to also be able to do common linear algebra operations  solving systems  least squares regression  decomposition  etc   look into LAPACK

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What about GLM   It s  based on the OpenGL Shading Language  GLSL  specification and released under the MIT license  Clearly aimed at graphics programmers

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I found this library quite simple and functional  http   kirillsprograms com top Vectors php   These are bare bone vectors implemented via C   templates  No fancy stuff - just what you need to do with vectors  add  subtract multiply  dot  etc

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If you are looking for high performance matrix linear algebra optimization on Intel processors  I d look at Intel s MKL library     MKL is carefully optimized for fast run-time performance - much of it based on the very mature BLAS LAPACK fortran standards   And its performance scales with the number of cores available   Hands-free scalability with available cores is the future of computing and I wouldn t use any math library for a new project doesn t support multi-core processors   Very briefly  it includes     Basic vector-vector  vector-matrix  and matrix-matrix operations  Matrix factorization  LU decomp  hermitian sparse   Least squares fitting and eigenvalue problems  Sparse linear system solvers Non-linear least squares solver  trust regions  Plus signal processing routines such as FFT and convolution Very fast random number generators  mersenne twist  Much more     see  link text   A downside is that the MKL API can be quite complex depending on the routines that you need   You could also take a look at their IPP  Integrated Performance Primitives  library which is geared toward high performance image processing operations  but is nevertheless quite broad   Paul   CenterSpace Software   NET Math libraries  centerspace net

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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

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FLENS  http   flens sf net  It also implements a lot of LAPACK functions

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I ll add vote for Eigen  I ported a lot of code  3D geometry  linear algebra and differential equations  from different libraries to this one - improving both performance and code readability in almost all cases    One advantage that wasn t mentioned  it s very easy to use SSE with Eigen  which significantly improves performance of 2D-3D operations  where everything can be padded to 128 bits

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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

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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

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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     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

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

Eigen2/Eigen3

GMTL

NT2

LAPACK & BLAS

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