# Two Taylor Algorithms for Computing the Action of the Matrix Exponential on a Vector
## 3 Numerical Experiments
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **Min.** & **Max.** & **Mean** & **Median** & **Std. Dev.** \\ \hline expmvtay1 & \(1.23\times 10^{-16}\) & \(1.95\times 10^{-14}\) & \(5.22\times 10^{-15}\) & \(3.75\times 10^{-15}\) & \(4.42\times 10^{-15}\) \\ expmvtay2 & \(1.23\times 10^{-16}\) & \(1.78\times 10^{-14}\) & \(5.46\times 10^{-15}\) & \(4.76\times 10^{-15}\) & \(4.35\times 10^{-15}\) \\ expmAvtay & \(1.70\times 10^{-16}\) & \(2.27\times 10^{-14}\) & \(7.41\times 10^{-15}\) & \(6.54\times 10^{-15}\) & \(5.46\times 10^{-15}\) \\ expmv & \(1.52\times 10^{-16}\) & \(2.62\times 10^{-14}\) & \(6.80\times 10^{-15}\) & \(4.98\times 10^{-15}\) & \(5.55\times 10^{-15}\) \\ expm_newAv & \(4.54\times 10^{-16}\) & \(1.63\times 10^{-13}\) & \(3.51\times 10^{-14}\) & \(2.68\times 10^{-14}\) & \(3.16\times 10^{-14}\) \\ expleja & \(1.58\times 10^{-16}\) & \(2.15\times 10^{-13}\) & \(4.37\times 10^{-14}\) & \(2.78\times 10^{-14}\) & \(4.69\times 10^{-14}\) \\ expv & \(1.55\times 10^{-16}\) & \(6.83\times 10^{-12}\) & \(4.09\times 10^{-13}\) & \(5.79\times 10^{-14}\) & \(1.01\times 10^{-12}\) \\ \hline expmvtay1 & \(5.34\times 10^{-16}\) & \(1.72\times 10^{-14}\) & \(5.55\times 10^{-15}\) & \(4.07\times 10^{-15}\) & \(4.40\times 10^{-15}\) \\ expmvtay2 & \(5.34\times 10^{-16}\) & \(3.08\times 10^{-14}\) & \(5.49\times 10^{-15}\) & \(3.87\times 10^{-15}\) & \(4.89\times 10^{-15}\) \\ expmAvtay & \(5.15\times 10^{-16}\) & \(2.71\times 10^{-14}\) & \(6.45\times 10^{-15}\) & \(5.85\times 10^{-15}\) & \(4.87\times 10^{-15}\) \\ expmv & \(4.04\times 10^{-16}\) & \(2.45\times 10^{-14}\) & \(6.57\times 10^{-15}\) & \(5.05\times 10^{-15}\) & \(5.41\times 10^{-15}\) \\ expm_newAv & \(9.63\times 10^{-16}\) & \(3.79\times 10^{-14}\) & \(1.02\times 10^{-14}\) & \(7.91\times 10^{-15}\) & \(8.30\times 10^{-15}\) \\ expjeja & \(3.68\times 10^{-16}\) & \(1.98\times 10^{-13}\) & \(4.22\times 10^{-14}\) & \(2.21\times 10^{-14}\) & \(5.03\times 10^{-14}\) \\ expv & \(6.13\times 10^{-16}\) & \(8.42\times 10^{-10}\) & \(1.58\times 10^{-11}\) & \(8.64\times 10^{-14}\) & \(9.70\times 10^{-11}\) \\ \hline expmvtay1 & \(1.81\times 10^{-16}\) & \(9.12\times 10^{-9}\) & \(2.17\times 10^{-10}\) & \(3.64\times 10^{-16}\) & \(1.41\times 10^{-9}\) \\ expmvtay2 & \(1.82\times 10^{-16}\) & \(9.12\times 10^{-9}\) & \(2.17\times 10^{-10}\) & \(3.76\times 10^{-16}\) & \(1.41\times 10^{-9}\) \\ expmAvtay & \(1.25\times 10^{-16}\) & \(4.94\times 10^{-9}\) & \(1.18\times 10^{-10}\) & \(5.04\times 10^{-16}\) & \(7.62\times 10^{-10}\) \\ expmv & \(1.75\times 10^{-16}\) & \(2.94\times 10^{-9}\) & \(7.01\times 10^{-11}\) & \(3.47\times 10^{-16}\) & \(4.54\times 10^{-10}\) \\ expm_newAv & \(1.25\times 10^{-16}\) & \(1.44\times 10^{-8}\) & \(3.44\times 10^{-10}\) & \(9.88\times 10^{-16}\) & \(2.23\times 10^{-9}\) \\ expleja & \(1.80\times 10^{-16}\) & \(1.45\times 10^{-6}\) & \(3.50\times 10^{-8}\) & \(1.05\times 10^{-15}\) & \(2.23\times 10^{-7}\) \\ expv & \(1.47\times 10^{-16}\) & \(9.43\times 10^{-7}\) & \(2.25\times 10^{-8}\) & \(1.27\times 10^{-15}\) & \(1.46\times 10^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Minimum, maximum, mean, median, and standard deviation values of the relative errors committed by all the codes for Sets 1, 2, and 3, respectively. For Set 3, this percentage dropped to 40%, followed in a 29% by expaja and in a 24% by expm_newAv. Table 4 compares the minimum, maximum, mean, and median values of the tuple \(m\) and \(s\), i.e., the order of the Taylor approximation polynomial and the value of the scaling parameter used by the first four methods. For the other codes, the parameter \(m\) is not comparable as it represents the degree of the Pade approximants to the matrix exponential (expm_newAv), the selected degree of interpolation (expjeja) or the dimension of the Krylov subspace employed (expmv). Additionally, \(s\) denotes the scaling value (expm_newAv) or the scaling steps (expjeja), but it is not provided for expv, because this code does not work with the scaling technique.