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.

Advertisements

Leave a Comment »

No comments yet.

RSS feed for comments on this post. TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Blog at WordPress.com.

%d bloggers like this: