A faster method of computing matrix pythagorean sums

In a recent paper [1], Moler and Morrison have described an iterative algorithm for the computation of the Pythagorean sum
a \oplus b {\underline {\underline \Delta} } (a^{2} + b^{2})^{1/2}
of two real numbers
a
and
b
without computing their squares or taking a square root. The subroutine is robust, short, portable, has a cubic rate of convergence, and is immune to floating-point overflows. In this note the method is extended to the efficient computation of the Pythagorean sum
A \oplus B
of two real commuting matrices.