Extension of accurate numerical algorithms for matrix multiplication based on error-free transformation

The error-free transformation of matrix multiplication is a useful technique for accurate numerical computations in linear algebra problems. It can be used to transform the product of two floating-point matrices into an unevaluated sum of floating-point matrices, making it useful for developing accurate numerical algorithms for matrix multiplication. This technique splits both left and right matrices into k floating-point matrices, and then \(\frac{1}{2}k(k+1)\) $\frac{1}{2}k(k+1)$ times matrix multiplications are performed. We extend this technique and propose several accurate algorithms for matrix multiplication, which involve p times matrix multiplications with \(p = 4, 5, 8, 9\) $p=4,5,8,9$ , respectively. The proposed algorithms efficiently provide more accurate results than those by double-precision arithmetic and less accurate than those by quadruple-precision arithmetic. In addition, we propose alternative forms to reduce the number of matrix multiplications with rounding errors. Numerical results show that the number of matrix multiplications affects the accuracy of the computed results. This dependence is examined using rounding error analysis and confirmed through numerical experiments.