How to approximate $\exp(-t(A-B))$ for diagonal $A$ and rank-1 $B$?

Question

I have $d\times d$ diagonal matrix $A=\operatorname{diag}a$ and rank-1 $B=b b^T$ and need to efficiently approximate the following function: $$f(t)=\langle\exp(-t(A-B))\rangle$$

Here $\langle M \rangle$ represents the sum of all entries in matrix $M$ and $0<t<d^2$. In my application, $a=2b(1-b)$ and $b$ is some vector of positive numbers adding up to 1.

My original solution of $f(t)\approx \langle\exp(-t A)\rangle$ was not accurate enough. Any advice what expansion I should use here?

Answer 1

It’s not super clear what you’re trying to do but you can use the Zassenhaus formula will give $$ e^{-t(A-B)}\approx e^{-tA} e^{tB} e^{t^2[A,B]}\ldots $$ There is obviously further simplification if $[A,B]=0$ or if $[A,[A,B]]=0$ or $[B,[A,B]]=0$ but that’s not clear if this is your case.

The link gives terms up to $t^4$ but this paper gives a systematic way of finding even higher order corrections.

Finally, the paper by

Casas, F., Murua, A., & Nadinic, M. (2012). Efficient computation of the Zassenhaus formula. Computer Physics Communications, 183(11), 2386-2391.

promises an efficient numerical implementation to compute higher order terms.