Relationship between taking a square root of a number and calculating
it via continued fraction (or, in a more natural setting, between
solving quadratic algebraic equation by closed formula and by
iterations) is a source of fascination of the present authors who do
so with matrices, mainly of small dimensions. They find the necessary
square roots in the explicit definition of the geometric mean of two
matrices, and have a lot of fun with proving convergence of their
matrix continued fractions and with demonstrating its quick rate on a
number of illustrative numerical examples.

CNO: 1948432 Raissouli, Mustapha ; Leazizi, Fatima .
Continued fraction expansion of the geometric matrix mean and applications.
Linear Algebra Appl. 359 (2003), 37--57.