Matrix-by-matrix multiplication algorithm with $O(N^2log_2N)$ computational complexity for variable precision arithmetic

Abstract

We show that assuming the availability of the processor with variable precision arithmetic, we can compute matrix-by-matrix multiplications in $O(N^2log_2N)$ computational complexity. We replace the standard matrix-by-matrix multiplications $\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22}\end{bmatrix}\begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22}\end{bmatrix}=\begin{bmatrix} A_{11}B_{11}+A_{12}B_{21} & A_{11}B_{12}+A_{12}B_{22} \\ A_{21}B_{11}+A_{22}B_{21} & A_{21}B_{12}+A_{22}B_{22}\end{bmatrix}$ by $\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22}\end{bmatrix}\begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22}\end{bmatrix}=\Bigl\lfloor\begin{bmatrix} (A_{11}+\epsilon A_{12})(B_{11}+1/{\epsilon}B_{21}) & (A_{11}+\epsilon A_{12})(B_{12}+1/{\epsilon}B_{22}) \\ (A_{21}+\epsilon A_{22})(B_{11}+1/{\epsilon}B_{21}) &(A_{21}+\epsilon A_{22})(B_{12}+1/{\epsilon}B_{22})\end{bmatrix} \Bigr\rfloor \% \frac{1}{\epsilon}$ where $\lfloor \rfloor$ denotes the floor, and $\%$ denotes the modulo operators. We reduce the number of block matrix-by-matrix multiplications from 8 to 4, keeping the number of additions equal to 4, and additionally introducing 4 multiplications of a block matrices by $\epsilon$ or $\frac{1}{\epsilon}$, and 4 floor and 4 modulo operations. The resulting computational complexity for two matrices of size $N\times N$ can be estimated from recursive equation $T(N)=4(N/2)^2$ (multiplication of a matrix by $\epsilon$ and $1/\epsilon$) plus $4(N/2)^2$ (additions of two matrices) plus $2N^2$ (floor and modulo) plus $4T(N/2)$ (four recursive calls) as $O(N^2log_2N)$. These multiplications of blocks of a matrix by number scales like $O((N/2)^2)$. We also present a MATLAB code using \emph{vpa} variable precision arithmetic emulator that can multiply matrices of size $N\times N$ using $(4log_2N+1)N^2$ vpa operations. This emulator uses $O(N)$ digits to run our algorithm.