Exponentials numerical | Eikonal Blog

2010.01.26

Def: $\varphi(t):\equiv \frac{||e^{t(A+E}-e^{tA}||}{||e^{tA}||}$ .

Estimate:
$\varphi(t) \le t ||E|| e^{t(\mu(A)-\eta(A)+||E||)}$
where:

$\mu(A):\equiv \max\{\Re(\lambda)|\lambda \in \Sigma_{\frac12(A+A^\dagger)}\}$ (i.e. the largest eigenvalue of the matrix $\frac12(A+A^\dagger)$ , i.e. “the logarithmic norm” of $A$ ),
$\eta(A):\equiv \max\{\Re(\lambda)|\lambda \in \Sigma_A\}$ ,
$\mu(A) \ge \eta(A)$ .

Source: T1268 = Qin Sheng “Global Error Estimates for Exponential Splitting”; IMA Journal of Numerical Analysis (1993) 14, 27-56.