Eikonal Blog

2010.01.26

Error estimates for exponential splittings

Filed under: mathematics — Tags: , — sandokan65 @ 11:58

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.

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