Computing exponentials of essentially non-negative matrices entrywise to high relative accuracy

A real square matrix is said to be essentially non-negative if all of its off-diagonal entries are non-negative. It has recently been shown that the exponential of an essentially non-negative matrix is determined entrywise to high relative accuracy by its entries up to a condition number intrinsic to the exponential function (Numer. Math. 110 (2008), 393-403). Thus the smaller entries of the exponential may be computed to the same relative accuracy as the bigger entries. This paper develops algorithms to compute exponentials of essentially non-negative matrices entrywise to high relative accuracy.