# Eikonal Blog

## 2010.02.15

### Lutzky’s expansion

Filed under: mathematics — Tags: — sandokan65 @ 22:16

$\frac{d e^{z}}{dt} = e^z \left\{\frac{dz}{dt},\frac{e^z-1}{z}\right\}$

$e^{z(t)} :\equiv e^{x}e^{t y}$

• $z(t) = \sum_{l=0}^\infty z_l t^l$
• $z_0 = x$
• $z_1 = \left\{ y , \frac{x}{e^x-1}\right\} = y - \frac1 [y,x] + \frac1{12} \{y,x^2\} - \frac1{720} \{y,x^4\} + \cdots$

Here the expansion $\frac{x}{e^x-1} = \sum_{k=0}^\infty \frac{B_k}{k!} x^k$ was used.

## Calculating $z_2$

Definition:

• $\frac{x e^{\xi x}}{e^x-1} = \sum_{k=0}^\infty \varphi_k(\xi) x^k$
• $\frac{e^{\xi x} -1}{e^x-1} = \sum_{k=0}^\infty \Psi_k(\xi) x^k$
• $\varphi_k(\xi) = \frac1{k!} \sum_{m=0}^\infty \binom{k}{m} B_m \xi^{k-m}$
• $\partial_\xi \varphi_{k+1}(\xi) = \varphi_k(\xi)$
• $\varphi_k(0)= \frac1{k!}B_k$,
• $\varphi_k(1)= \varphi_k(0)$ (for $k\ne 1$),
• $\varphi_0(\xi) \equiv 1$
• $\int_0^1 \varphi_k(\xi) d\xi = \delta_{k,0}$,
• $\Psi_{k}(\xi) = \varphi_{k+1}(\xi) - \varphi_{k+1}(0)$,
• $\int_0^1 \varphi_k(\xi) \Psi_{l}(\xi) d\xi = \delta_{k,0}\varphi_{l+1}(0) \theta(l) + \delta_{l,0}\varphi_{k+1}(0) \theta(k)$,

Now:
$z_2 = - \sum_{l=1}^\infty \sum_{k=0}^\infty \{y,x^l,y,x^k\} \frac{B_{l+1}B_k}{(l+1)!k!} -\frac12 \sum_{k=1}^\infty \sum_{l=1}^\infty \sum_{j=0}^\infty I_{kl} \{\Delta_{kl}, x^j\} \frac{B_j}{j!}$
where:

• $\{y,x^k,y\} :\equiv [\{y,x^k\},y]$,
• $\{y,x^l,y,x^k\} :\equiv \{\{y,x^l,y\},x^k\}$,
• $\Delta_{kl} :\equiv [\{y,x^l,\{y,x^k\}] = - \Delta_{lk}$,
• $I_{kl} :\equiv \int_0^1 \varphi_k(\xi) \varphi_{l+1}(\xi)$, ($I_{l+2,l}=0$)

The more compact expression is:
$z_2 = -\frac12 \left\{ J, \frac{x}{e^x-1}\right\}$
where
$J :\equiv \int_0^1 d\xi \left[\{z_1,e^{x\xi}\}, \left\{z_1,\frac{e^{x\xi}-1}{x}\right\}\right] = \int_0^1 d\xi \left[\{y,\frac{xe^{x\xi}}{e^x-1}\}, \left\{y,\frac{e^{x\xi}-1}{e^{x}-1}\right\}\right].$

The Hausdorff recursion formula:
$z_n = \frac1{n} \left(\left\{y, \frac{x}{e^x-1}\right\} D_x \right) z_{n-1}$.

Source: T1265 = M. Lutzky “Parameter Differentiation of Exponential Operators and the Baker-Campbell-Hausdorff Formula”; Journal of Mathematics Physics, Vol 9, N 7, July 1968.

## 2010.01.28

### BCS (Baker–Campbell–Hausdorff) formula

Filed under: Uncategorized — Tags: , — sandokan65 @ 17:12

In this posting the matrix $C(t)$ is defined by $e^{C(t)}:\equiv e^{t(A+B)}$ where $A$ and $B$ are constant matrices.

The Baker–Campbell–Hausdorff theorem claims that:

$C(t) = B + \int_0^1 dt g(e^{t a} e^b ) A$,

where $g(z):\equiv \frac{\ln(z)}{z-1} = \sum_{m=0}^\infty \frac{(1-z)^m}{m+1}$, and the lower-case letters $a$ and $b$ represent the adjoint actions of the corresponding matrices $A$ and $B$ (e.g. $a X:\equiv ad(A) X :\equiv [A,X]$).

One is frequently seeing the following series expression:

$C(t) = t (A+B) + \frac{t^2}2 [A,B] + \frac{t^3}{12} ([[A.B],B]-[[A,B],A]) + \cdots = t (A+B) + \frac{t^2}2 a B + \frac{t^3}{12} (b^2 A + a^2 B) + \cdots$

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

### The Trotter Formula

Filed under: mathematics — Tags: — sandokan65 @ 17:02

$e^{t(A+B)} = \lim_{m\rightarrow \infty} \left(e^{\frac{t}{m}A}e^{\frac{t}{m}B}\right)^m$.

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

### The Zassenhaus formula

Filed under: mathematics — Tags: , — sandokan65 @ 16:49

$e^{A+B} = e^{A}e^{B}e^{C_2}e^{C_3}\cdots$
where

• $C_2 = -\frac12 [A,B] = -\frac12 a B$,
• $C_3 = -\frac13 [[A,B],B] - \frac16 [[A,B],A] = -\frac13 b^2 A + \frac16 a^2 B$, etc.

Here, as elsewhere on this site, I am using notation $a:\equiv ad(A)$ etc.

Sources:

• T1264 = W. Magnus “On the exponential solution of differential equations for a linear operator”; Communications of Pure and Applied mathematcis, Vol VII, 6490673 (1954).
• T1269 = A.N.Richmond “Expansion for the exponential of a sum of matrices”; Int. Journal of Control (issue and year unknown).

## 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/

### 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).