Eikonal Blog

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).

2010.01.22

BCS formula on computer

Filed under: mathematics — Tags: , , — sandokan65 @ 16:06

Def: e^C :\equiv e^A e^B

C = A + \int_0^1 dt \psi(e^{a}e^{t b}) B,
where a:\equiv ad(A), b:\equiv ad(B) and
\psi(z):\equiv \frac{z \ln(z)}{z-1} = 1 + \frac{w}{1\cdot 2} - \frac{w^2}{2\cdot 3} + \frac{w^3}{3\cdot 4} - \cdots for w:\equiv z-1.

Richtmyer and Greenspan obtained the 512 initial terms of that expansion, but here I present only several of them:

C = A + B + \frac12 a B + \frac1{12} a^2 B - \frac1{12} b a B + \frac1{12,096} a^4 b^2 a B - \frac1{6,048} a^3 b a^3 B - \frac1{3,780} a^3 b a b a B + ...

Not all terms are independent, due to Jacobi identities.

Sources:

  • T1267: R.D. Richtmyer and S. Greenspan: “Expansion of the Campbell-Baker-Hausdorff formula by Computer”; Communications on Pure and Applied Mathematics, Vol XVIII 107-108 (1965).

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