English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

One of the most famous conjectures in computer algebra is that matrixmultiplication might be feasible in not much more than quadratic time. The bestknown exponent is 2.376, due to Coppersmith and Winograd. Many attempts tosolve this problems in the literature work by solving, fixed-size problems andthen apply the solution recursively. This leads to pure combinatorialoptimisation problems with fixed size. These problems are unlikely to besolvable in polynomial time.  In 1976 Laderman published a method to multiply two 3x3 matrices using only23 multiplications. This result is non-commutative, and therefore can beapplied recursively to smaller sub-matrices. In 35 years nobody was able to dobetter and it remains an open problem if this can be done with 22multiplications. We proceed by solving the so called Brent equations [7]. Wehave implemented a method to converting this very hard problem to a SATproblem, and we have attempted to solve it, with our portfolio of some 500 SATsolvers. With this new method we were able to produce new solutions to theLaderman's problem. We present a new fully general non-commutative solutionwith 23 multiplications and show that this solution is new and is NOT anequivalent variant of the Laderman's original solution. This resultdemonstrates that the space of solutions to Laderman's problem is larger thanexpected, and therefore it becomes now more plausible that a solution with 22multiplications exists. If it exists, we might be able to find it soon just byrunning our algorithms longer, or due to further improvements in the SAT solveralgorithms.

A New General-Purpose Method to Multiply 3x3 Matrices Using Only 23 Multiplications