# Eikonal Blog

## 2010.01.26

### Magnus on matrix exponentials

Definition: The Magnus’ bracket is defined as $\{y,x^n\}_M :\equiv (- ad(x))^n y$.

Properties:

• $e^{-ad(x)} y = \sum_{n=0}^\infty \{y, x^n\}_M$
• for a power series $P(x)=\sum_{n=0}^\infty p_l x^l$ one has $\{y, P(x)\}_M = \sum_{n=0}^\infty p_l \{y, x^n\}_M$
• Hausdorf:
• $e^{-x} (y D_x) e^x = \left\{y,\frac{e^x -1}{x}\right\}_M$,
• $(y D_x e^{-x}) e^x = \left\{y,\frac{1-e^{-x}}{x}\right\}_M$,

where the Hausdorff polarization operator $(y D_x)$ is defined as following:

$(y D_x) x^n :\equiv \sum_{k=0}^{n-1} x^k y x^{n-k-1}$.

• Lemma: For two power series $P(x)$ and $Q(x)$ s/t $P(x) Q(x) = 1$ one has the following equivalence:
$\{y,P(x)\}_M=u \Leftrightarrow y = \{u, Q(x)\}_M$.
• $\{\{y,x^n\}_M,x^m\}_M = \{y,x^{n+m}\}_M$.

For $e^{z} :\equiv e^{x}e^{y}$ one has:

• $z=\ln(1+u) = u - \frac12 u^2 + \frac13 u^3 + \cdots$

where
$u:\equiv e^x e^y - 1 = x + y + \frac12 x^2 + xy + \frac12 y^2 + \cdots$
• $z = x + y + \frac12 [x,y] + \frac1{12} \{x,y^2\}_M + \frac1{12} \{y,x^2\}_M -\frac1{24} [\{x,y^2\}_M,x] - \frac1{720} \{x,y^4\}_M - \frac1{720} \{y,x^4\}_M + \frac1{180} [[\Delta_1,x],y] - \frac1{180}\{\Delta_1,y^2\}_M - \frac1{120} [\Delta_1,\Delta] - \frac1{360} [\Delta_2,\Delta] + \cdots$

where

$\Delta=[x,y]$,
$\Delta_1=[\Delta,x]$,
$\Delta_2=[\Delta,y]$.

## The Magnus’ formula for ordered exponentials

And this is really beautiful thing: for the ordered exponential $U(t|A):\equiv \left(e^{\int_0^t dt_1 A(t_1)}\right)_+$ one can observe its logarithm (the Magnus function of A) $\Omega(t|A) :\equiv \ln U(t|A)$ (i.e. $e^{\Omega}=U$) and get following nonlinear ODE for it:

$\frac{d\Omega(t)}{dt} = \{A, \frac{\Omega}{1-e^{-\Omega}}\}_M = \sum_{n=0}^\infty \beta_n \{A,\Omega^n\}_M = A + \frac12 [A,\Omega] + \frac1{12}\{A,\Omega^2\}_M + \cdots$

where $\beta_{2n1+1}=0$ ($\forall n\in{\Bbb N}_0$), $\beta_{2n} = (-)^{n-1} \frac{B_{2n}}{(2n)!}$ and $B_n$‘s are ordinary Bernoulli’s numbers.

Source: T1264 = W. Magnus “On the exponential solution of differential equations for a linear operator”; Communications of Pure and Applied mathematics, Vol VII, 6490673 (1954). Note: This paper is probably mother of the whole this area of matrix exponentials.

Related here: Magnus’ Bracket Operator – https://eikonal.wordpress.com/2010/02/15/magnus-bracket-operator/

### 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: $\partial_z (z\partial_z {\cal F}(z)) = A{\cal F}(z)+{\cal F}(z)B$.

—-
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.