Table 1 Algorithm: How to calculate exponential matrix with PIM

1. Initial calculation

Choose time step τ and bisection order n.

2. Choose two approximation methods ofT(τ)

a. Fourth-order Taylor approximation

b. (4,4)-Padé approximation

3. For each discretized point, compute

for (o = 1:n) \({{\boldsymbol{T}}_{o}} = 2{{\boldsymbol{T}}_{o}} + {{\boldsymbol{T}}_{o}} \times{{\boldsymbol{T}}_{o}}\) end

4. Calculate the matrix exponential function

\({\boldsymbol{T}} ( \tau) = {\boldsymbol{I}} + {{\boldsymbol {T}}_{o}} = {e^{{\boldsymbol{H}}\tau}}\)