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They are globally and quadratically convergent, except for MARE in the critical case at which it converges linearly with the linear rate $1/2$. However, the initialization phase and the doubling iteration kernel of any doubling algorithm involve inverting nonsingular $M$-matrices. In particular for MARE in the critical case, the $M$-matrices in the doubling iteration kernel,

although nonsingular, move towards singular $M$-matrices at convergence. A nonsingular $M$-matrix can be inverted by the GTH-like algorithm to almost full entrywise relative accuracy, provided a triplet representation of the matrix is known. Recently, Nguyen and Poloni ({\em Numer. Math.}, 130(4):763--792, 2015) discovered a way to construct triplet representations in a cancellation-free manner for all involved $M$-matrices in the doubling iteration kernel, for

a special class of MAREs arising from Markov-modulated fluid queues.

In this paper, we extend Nguyen's and Poloni's work to all MAREs by also devising a way to construct the triplet representations cancellation-free.Our construction, however, is not a straightforward extension of theirs. It is made possible byan introduction of novel recursively computable auxiliary nonnegative vectors. As the second contribution, we propose an entrywise relative residual for an approximate solution. The residual has an appealing feature of being able to reveal the entrywise relative accuracies of all entries, large and small, of the approximation. This is in marked contrast to the usual legacy normalized residual which reflects relative accuracies of large entries well but not so much those of very tiny entries. Numerical examples are presented to demonstrate and confirm our claims.