Hi, [crossposted to oo-numerics and boost-developers] About 20 months back, I asked if there was any interest in a geometric or Clifford algebra library written in C++. Despite some positive replies, I got the impression that there were not a lot of people in the Boost community who saw any practical uses for such library. Back then I had a rudimentary proof-of-concept for a Clifford algebra library that used meta-programming techniques to gain significant performance benefits over other existing libraries (Gaigen, nKlein, CLU, etcetera). Over the past few months I have put a lot more work into this library. The bulk of the work consisted of adding more unit-tests, proving many of the theorems in some of the papers by David Hestenes et al. I also added expression templates to improve performance. To give an idea of what the library is capable of: #include <boost/math/clifford/clifford.hpp> #include <boost/mpl/list_c.hpp> #include <cassert> using namespace boost::math::clifford; using namespace boost::mpl; typedef algebra<4, 1> conformal_model; typedef blade<float, 0, conformal_model> scalar; typedef blade<float, 1, conformal_model> vector; typedef blade<float, 5, conformal_model> pseudo_scalar; typedef blade<float, 2, conformal_model> bi_vector; // same as: // typedef element<float, list_c<int, 1>, conformal_model> bi_vector; typedef list_c<int, 0, 2> spinor_grades; typedef element<float, spinor_grades, conformal_model> spinor; typedef list_c<int, 0, 1, 2, 3, 4, 5> multi_vector_grades; typedef element<float, multi_vector_grades, conformal_model> multi_vector; int main() { assert(sizeof(scalar) == sizeof(float)); assert(sizeof(vector) == 5 * sizeof(float)); assert(sizeof(bi_vector) == 10 * sizeof(float)); assert(sizeof(pseudo_scalar) == sizeof(float)); assert(sizeof(spinor) == 11 * sizeof(float)); assert(sizeof(multi_vector) == 32 * sizeof(float)); vector e_plus; e_plus[3] = 1.0; vector e_min; e_min[4] = 1.0; e_star = e_min - e_plus; e_star *= -std::sqrt(2.0); e_star /= 2.0f; e [noalias] = e_plus * std::sqrt(2.0) - e_star; bi_vector E = outer_product(e, e_star); // geometric product between bi_vector and vector multi_vector v = E * e_star; // geometric product between two vectors, generating a spinor vector a, b; spinor s = a * b; // doing a spinor rotation vector c; c [noalias] = reverse(s) * a * s; } Each product always does the bare minimum of additions, multiplications and subtractions to calculate the result on the left-hand-side of an expression. Thus, if you do the full geometric product of two multi_vectors but assign it to a scalar, it will only do those calculations that are involved in grade 0. Also, unlike other existing libraries, no dynamic allocations are being done ever. Currently, the following operations are supported, between any two random elements of an algebra (blades and composite multi_vectors): geometric_product = operator * inner_product outer_product fat_dot_product left_contraction right_contraction scalar_product of course, all the normal operations (scalar multiplication, addition of elements, etcetera) are supported, as well as reversion, involution and conjugates. Adding more operators is trivial. They can either be implemented in terms of existing operators, or by implementing an appropriate grade-predicate. For example, this is what the implementation of the outer_product looks like: struct outer_product_predicate { template <class ResultGrade, class LhsGrade, class RhsGrade> struct apply { typedef typename mpl::if_ < typename mpl::equal_to < ResultGrade, typename mpl::plus<LhsGrade,RhsGrade>::type >::type, mpl::true_, mpl::false_ >::type type; }; }; BOOST_MATH_CLIFFORD_PRODUCT_IMPL( outer_product, impl::outer_product_predicate ) which for somebody that knows the definition of the outer_product is easy to understand. I have unit-tests for 2, 3 4 and 5 dimensional algebras, but more is possible if your compiler supports it. Currently I have only tested the library on MSVC 7.1. I tried 7.0 but couldn't get it to work (excessive use of MPL). GCC and Intel compiled it at one point, but because I only use MSVC I dropped support. Of course, if anybody is interested I'd be trying other compilers again. You will need a compiler that does aggresive inlining though. On MSVC I am using the force_inline feature to get the release performance I need. Anyway, let me know if anybody is interested, and I'll put together a package to upload somewhere. Cheers, Jaap