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We propose splitting methods for the computation of the exponential ofperturbed matrices which can be written as the sum $A=D+\varepsilon B$ of asparse and efficiently exponentiable matrix $D$ with sparse exponential $e^D$and a dense matrix $\varepsilon B$ which is of small norm in comparison with$D$. The predominant algorithm is based on scaling the large matrix $A$ by asmall number $2^{-s}$, which is then exponentiated by efficient Pad\'e orTaylor methods and finally squared in order to obtain an approximation for thefull exponential. In this setting, the main portion of the computational costarises from dense-matrix multiplications and we present a modified squaringwhich takes advantage of the smallness of the perturbed matrix $B$ in order toreduce the number of squarings necessary. Theoretical results on local errorand error propagation for splitting methods are complemented with numericalexperiments and show a clear improvement over existing methods when mediumprecision is sought.

The Scaling, Splitting and Squaring Method for the Exponential of Perturbed Matrices