New Practical Strassen-like Approximate Matrix-multiplication Algorithms Found via Solving a System of Cubic Equations

We have encapsulated the concept of matrix multiplication into a system of cubic equations. We discuss the properties of this system and the symmetry group of the set of solutions. Using solutions of these equations, we propose new algorithms for matrix multiplication, with explicit inequalities that bound the error. Also we calculate exactly the “hidden coefficient” in the Big-Oh notation for this class of algorithms. Moreover, we present a fourstage process for solving this system of cubic equations numerically, which we believe is new. Specifically, by approximately solving such a system of these equations with 594 unknowns and 729 equations, we have found an approximate algorithm for multiplying 3× 3 matrices in 22 steps, as compared to 23 steps by Laderman, 24 steps by Hopcroft and Kerr, 25 steps by Gastinel, and 27 steps by the näıve algorithm. This gives rise to a matrix multiplication algorithm which runs in time n2.814··· as compared to n2.854···, n2.893···, n2.930···, and n3 respectively. However, those running times are all inferior to Strassen’s Algorithm, the currently fastest practical known method, which runs in time n2.807···. We have attempted to also solve these cubic equations in an instantiation with 4096 equations and 2304 or 2256 unknowns, a solution of which would give rise to an algorithm for approximate matrix multiplication which is practical and asymptotically faster than Strassen’s algorithm, by performing 4×4 matrix multiplication in 48 or 47 steps. Lastly, we have investigated an algorithm for performing 5 × 5 matrix multiplication in 99 steps, which would exceed the current record, but is less than the 91 required to beat Strassen’s algorithm. We do not have solutions to the 4× 4 or 5× 5 problems at this time. Furthermore, we have discovered that these equations were known to Richard Brent several years ago [Bre70], and accordingly we name them the Brent Equations. In addition, if these equations can be solved exactly, over any field, then it will produce an asymptotically-fast exact algorithm over that field and all its finite-degree algebraic extensions. We are unable to do that at this time.