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Alex E wrote:

"The algorithm of such calculations is contained in the book
of Meeus "Astronomical Algorithms". I doubt that these calculations
can be easily done by hand, but they can be done with a computer
and I suppose that Frank Reed's program can do this."

 

I've got software that implements those algorithms, but that's not necessarily the best way to do it these days, and it's not what my online almanac tools are doing. The JPL DE406 ephemeris data is publicly available and exceptionaly accurate. With some coding trickery it can be converted into a database about ten megabytes in size. From there, it's just a "lookup" and an interpolation to get the positions for any date and time (in the range of validity).

 

But to George's point, the Solar System has some moderate chaotic behavior on long time scales, and that implies that the computational cost increases exponentially. That makes it deterministic but in a useless manner. It's worth comparing this with the weather. Weather prediction is the textbook (or coffee-table book) case of chaotic dynamics in a system where prediction might be very helpful. The computational cost of a weather forecast increases exponentially, literally doubling for every day or two that you add to the forecast. So we can read a forecast in the newspaper for three days in the future and it may have some validity, but we will probably never have a weather forecast that can reliably predict whether it will be raining thirty days from today. The computational cost of suc a forecast would be astronomical. Some authors describing this chaotic behavior get a little carried away and write in despair that the future is hopelessly unpredictable. And yet, with only trivial computational cost, I can practically guarantee you that there will be one day in July of 2005 when the high temperature in Mystic, Connecticut will be within five degrees of 85 degrees Fahrenheit. Chaos excludes some aspects of predictability... but not all.