In this work we derive efficient rational L∞ approximations of various degrees for the quadruple precision computation of the matrix exponential. We focus especially on the two classes of normal and nonnegative matrices. Our method relies on Remez algorithm for rational approximation while the innovation here is the choice of the starting set of non-symmetrical Chebyshev points. Only one Remez iteration is then usually enough to quickly approach the actual L∞ approximant.
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