I need to approximate the following in $O(d)$ time for $d\times d$ diagonal $A$ and rank-1 $B$ $$u^T \exp(-A+B) v$$

Here $u,v$ are vectors in $\mathbb{R^+}^d$, $A,B$ are positive semi-definite and $B$ is relatively small

The following approximations take $O(d)$ to compute and get me within factor of 2 of true value on sample data

  • set $B$ to 0
  • truncate Zassenhaus formula at first term

However, adding more terms from the Zassenhaus expansion seems to make the approximation worse, any tips?


  • $\begingroup$ How is your title correct, what is the matrix that receives a rank-1 update? $\endgroup$ Mar 18 at 12:55
  • $\begingroup$ the argument of the matrix function. IE, I can compute $f(X)$ cheaply, and need to compute $f(X+rank1)$ where $f(X)=u^T e^X v$ $\endgroup$ Mar 18 at 16:07

1 Answer 1


There is work on low-rank updates of matrix functions, for instance this one:

Beckermann, Bernhard; Kressner, Daniel; Schweitzer, Marcel, Low-rank updates of matrix functions, SIAM J. Matrix Anal. Appl. 39, No. 1, 539-565 (2018). ZBL1390.15024.

It could be a useful starting point if you wish to explore in that direction.


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