A search for the specific coefficients listed led me to the method ROS3PRL from
J. Sieber, Konvergenzanalyse und Numerische Tests für die Prothero–Robinson–Gleichung (Master thesis), TU Darmstadt, 2014.
I can't seem to find this thesis online, but the method is mentioned in the following which may be of interest.
Rang, Joachim. "Improved traditional ...