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I had a quick read at this paper. If I understand correctly, that's concerned with finding a rigorous enclosure of the exponential of a floating point matrix, not of a general interval matrix?
That being said, the INTLAB methods mentioned in the paper (verifylss and verifyeig) are both already implemented in ILA.jl,
https://epubs.siam.org/doi/abs/10.1137/19M1263431
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