Consider the system of linear equations:
$$ Ax=b \tag{1} \label{eq1} $$
where
- $A\in\mathbb F^{n\times n}$, diagonalizable dense matrix, over the field $\mathbb F$ of real or complex numbers,
- $x\in \mathbb F^{n\times 1}$ is a vector of unknowns
- $b\in \mathbb F^{n\times 1}$ is a known right-hand side vector
- $n$ is in the order of 1000–10000
Unlike the usual system of linear equations, I do not have access to $A$ itself; however, I have access to the matrix exponential $e^A$. The matrix exponential is accessible both as an explicit matrix and, consequently, as a function acting on a vector.
What are my options for finding the solution of $\eqref{eq1}$ knowing only the matrix exponential? Computing matrix logarithm does not seem the best option as I doubt being able to get anything numerically stable and reasonably efficient.
Am I missing something simple?
logm(E) \ b
, before writing it off as unstable / inefficient? 2. Are $A$ and $E=\exp(A)$ sparse or dense? $\endgroup$