Abstract
This paper is concerned with a relative perturbation theory and its entrywise relatively accurate numerical solutions of an M-matrix Sylvester equation AX + XB = C by which we mean both A and B have positive diagonal entries and nonpositive off-diagonal entries and \({P=I_m \otimes A+B^{\rm T} \otimes I_n}\) is a nonsingular M-matrix, and C is entrywise nonnegative. It is proved that small relative perturbations to the entries of A, B, and C introduce small relative errors to the entries of the solution X. Thus the smaller entries of X do not suffer bigger relative errors than its larger entries, unlikely the existing perturbation theory for (general) Sylvester equations. We then discuss some minor but crucial implementation changes to three existing numerical methods so that they can be used to compute X as accurately as the input data deserve.
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Xue, J., Xu, S. & Li, RC. Accurate solutions of M-matrix Sylvester equations. Numer. Math. 120, 639–670 (2012). https://doi.org/10.1007/s00211-011-0420-1
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DOI: https://doi.org/10.1007/s00211-011-0420-1