[QUESTION] How to compute a Jacobian matrix efficiently using automatic differentiation

[Yidong-ZHAO]

Hi!

I have a multi-valued function $\bf{f}: \mathbb{R}^n \to \mathbb{R}^n$, where $n$ indicates the dimension of the input/output vector. The naive way to get the Jacobian matrix $\bf{J}$ of $\bf{f}$ is to compute the components rows by rows using automatic differentiation (following the official document).

However, for a large $n$, the naive way is very slow. I wonder whether there is an efficient way similar to what is suggested in this discussion.

In my case, for each row $i \in [0, n)$, $J$ has a maximum of 9 non-zero components. Their column indices are: $i-m-1$, $i-m$, $i-m+1$, $i-1$, $i$, $i+1$, $i+m-1$, $i+m$, $i+m+1$, where $m \in (0, n)$ is a constant.

Thanks!