An $O(n^2{\log }^{4}n\log \log n)$ Time Matrix Multiplication Algorithm

We show, for the input vectors $(a_0,a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_{n-1})$ and $(b_0, b_1, …, b_{n-1})$,where$a_i${}^{\mathrm{\prime }}sand$b_j${}^{\mathrm{\prime }}sarerealnumbers,after$O(n\log^4 n)$timepreprocessingforeachofthem,thevectormultiplication$(a_0, a_1, …, a_{n-1})(b_0, b_1, …, b_{n-1})^T $canbecomputedin$O(\log^4 n \log \log n)$time\mathrm{.}Thisenablesthematrixmultiplicationfortwo$n\times n$matricestobecomputedin$O(n^2 \log^4 n\log \log n)$ time.