If you guys want to know how prediction of the eclipses work, open any astronomy textbook or consult any astronomy source on the topic. That's how it's predicted. It's all there. If you are curious about the details, look into it. It has nothing to do with the Wiki.
Here: http://eclipsewise.com/help/de405-predictions.html
The coordinates of the Sun used in these eclipse predictions have been calculated on the basis of the JPL DE405 solar system ephemeris. This ephemeris consists of computer representations of the positions, velocities and accelerations of major Solar System bodies, tabulated at equally spaced intervals of time, covering the span 1599 Dec 09 to 2201 Feb 20. Beginning in 2003, the JPL DE405 has been the basis of the Astronomical Almanac. See Jet Propulsion Laboratory Developmental Ephemeris for more information of the JPL ephemerides.
So not using the Saros cycle like you say it is.
It says that the model was only used to get the position of the sun, not to predict the eclipse.
Do some searching and you will find that JPL DE405 is based on perturbation prediction.
It models the positions of the Sun, the Moon, and every known planet, using Newtonian gravity, general relativity, and some tidal-force effects studied. This information is one link away from the page I linked.
You're intent on driving this "perturbation" point for every simulation that has ever been used, yet you very clearly either didn't actually do those searches you're talking about, or cherrypicked what you found. If you show me some sources and provide quotes for debate then I'll take the claim seriously.