Numerical methods for computing exponential, if I have computed an exponential of a perturbated matrix

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I need to compute the product $e^{H_1}\,e^{H_2}\,\ldots\,e^{H_n}$ for antihermitian matrices $H_j$ that do not commute and $H_i-H_{i+1}$ is small. Is there a numerically convenient way to compute $e^{H_{i+1}}$ if I have computed (using some methods you might recommend) $e^{H_i}$? Is there anything I might use from the previous computation?