INTRODUCTION ============ M4RI is a library for fast arithmetic with dense matrices over F2. The name M4RI comes from the first implemented algorithm: The "Method of the Four Russian"” inversion algorithm published by Gregory Bard. This algorithm in turn is named after the "Method of the Four Russians" multiplication algorithm which is probably better referred to as Kronrod's method. M4RI is available at http://m4ri.sagemath.org FEATURES ======== * basic arithmetic with dense matrices over F2 (addition, equality testing, stacking, augmenting, sub-matrices, randomisation); * asymptotically fast O(n^log_2(7)) matrix multiplication via the "Method of the Four Russians" (M4RM) & Strassen-Winograd algorithm; * fast row echelon form computation and matrix inversion via the "Method of the Four Russians" (M4RI, O(n^3/log(n))); * asymptotically fast PLUQ factorisation; * support for the x86/x86_64 SSE2 instruction set where available; * preliminary support for parallisation on shared memory systems via OpenMP; * and support for Linux and OS X (GCC), support for Solaris (Sun Studio Express) and support for Windows (Visual Studio 2008 Express). OPENMP SUPPORT ============== OpenMP support for parallel multiplication and elimination is enabled with the --with-openmp configure switch. If GCC is used to compile the library it is avised to use at least GCC 4.3 since earlier versions have problems with OpenMP in shared libraries. OpenMP support was introduced in GCC 4.2. Both MSVC and SunCC support OpenMP but we have no experience with these yet. Generally speaking better performance improvements can be expected on dual-core Opteron CPUs than on dual-core Core2Duo CPUs. This is because the later has a shared L2 cache which is already almost fully utilised in the single-core implementation. Overall, the speed-up is considerably but sublinear. See http://m4ri.sagemath.org/performance.html for details.