Preservation of positive-diagonal H-matrices by scaled Newton and Halley square-root iterations

Every scaled Newton square-root iterate of a real nonsingular H-matrix with
positive diagonal entries is again a real nonsingular H-matrix with positive
diagonal entries, when initialized by a positive scalar multiple of the
identity, a positive scalar multiple of the input, or more generally a nonzero
nonnegative affine function of the input. The scaling parameter may be any
positive scalar at each step. No normalization of the input diagonal is
required. The same conclusions hold for scaled Halley square-root iteration. A
single positive diagonal similarity makes all the iterates strictly row
diagonally dominant. The proof uses an elementary class of rational functions
closed under positive linear combinations and the transformation
$f(z) \mapsto z/f(z)$, together with a resolvent comparison inequality. The
theorem addresses the stated initialization and positive-diagonal hypotheses;
it is not a result for arbitrary commuting H-matrix initializations.