Eikonal Blog

2010.01.26

A Theorem of Ellis and Pinsky

Filed under: mathematics — Tags: — sandokan65 @ 13:12

Theorem (Ellis & Pinsky): Let $\latex A$, B be real symmetric n\times n matrices, and assume that B is negative semi-definite. Then as \epsilon \rightarrow +0:

  • e^{t(A+i\frac1{\epsilon}B)} e^{-i\frac{t}{\epsilon}B} = e^{t A_0} + {\cal O}(\epsilon),

where A_0 :\equiv \sum_{\lambda\in\sigma(B)} P_\lambda A P_\lambda, here P_\lambda being the orthogonal projector on the eigenspace of B associated with the eigenvalue \lambda.

Source: T1270 = S.L.Campbell “On the limit of a product of matrix exponentials”; Linear and Multilinear Algebra; 1978, Vol 6, p.p. 55-59.

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Expansions of the exponentials of the sums of matrices

Richmond’s formula #1: e^{t(A+B)} = e^{tA}e^{tB} + \sum_{r=0}^\infty E_r t^r, where:

  • E_{r+1} = \frac1{r+1}\left((A+B)E_r+[B,F_r]\right) \ with E_0=E_1=0,
  • F_{r+1} = \frac1{r+1}(A F_r + F_r B) \ with F_0=1.

Richmond’s formula #2: e^{t(A+B)} = \frac12(e^{tA}e^{tB}+e^{tB}e^{tA}) + \sum_{r=0}^\infty E'_r t^r, where:

  • E'_{f+1} = \frac1{r+1}((A+B)E'_r+\frac12[B,F_r]+\frac12[A,F^{*}_r]) \ with E'_0=E'_1=E'_2=0, and where F_r‘s are the same as in the Richmond’s formula #1.
  • Following bounds are valid: ||E'_r|| < \frac1{(r-1)!} (||A||+||B||)^r.

Note that the exponential generating function {\cal F}(z):\equiv \sum_{r=0}^\infty \frac{z^r}{r!} F_r satisfies the second order ODE: \partial_z (z\partial_z {\cal F}(z)) = A{\cal F}(z)+{\cal F}(z)B.

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Source: T1269 = A.N.Richmond “Expansion for the exponential of a sum of matrices”; Int. Journal of Control (issue and year unknown).

Exponential splittings

Filed under: mathematics — Tags: — sandokan65 @ 12:06

First order splitting:

  • e^{t\sum_{i=1}^{n} A_i } = \prod_{i=1}^n e^{t A_i} + {\cal O}(t^2).

Second order splittings:

  • The Strang’s Splitting:
    e^{t\sum_{i=1}^{n} A_i } = e^{\frac{t}2 A_1} e^{\frac{t}2 A_2} \cdots e^{\frac{t}2 A_{n-1}} e^{t A_n} e^{\frac{t}2 A_{n-1}} \cdots e^{\frac{t}2 A_2} e^{\frac{t}2 A_1} + {\cal O}(t^3).
  • The parallel Splitting:
    e^{t\sum_{i=1}^{n} A_i } = \frac12(e^{t A_1}\cdots e^{t A_n} + e^{t A_n}\cdots e^{t A_1})  + {\cal O}(t^3).

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

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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