# 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.

### 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.