🚀 go-pugleaf

I am interested in using a symmetric method to integrate a second
order differential equation in time (we are integrating the
equations of motion of each of the planets in the solar system).
The symmetric method can be written:

  x(n+1) = - sum(j=1 to k) a(k-j) x(n-j+1) +
                   h**2 sum(j=1 to k) b(k-j) f(n-j+1)

where we are solving x''(t) = f(x,t), x(n) is the solution
on the nth time step, and a and b are coefficients (reference:
"A Three Million Year Integration of the Earth's Orbit",
T. R. Quinn, S. Tremaine, and M. Duncan, The Astronomical
Journal, vol 101, no 4, June 1991, pages 2287-2305).
We are interested in reproducing some of the results in
the paper I listed as the reference.  They used k=12,
which means that the coefficients a and b were chosen
to that the method integrates a 13th order polynomial
exactly.  Note, a(j) = a(k-j), b(j) = b(k-j), a(0) = 1,
and b(0) = 1.  So, there are 12 coefficients I need.
Obviously I can work out these coefficients from scratch,
but if anyone happens to have them handy, you could save me
alot of effort if you could kindly send me a list.  Many
thanks!

John
-- 
Dr. John K. Prentice                      
Partner, Quetzal Computational Associates    
3200 Carlisle N.E., Albuquerque, NM   87110-1664
Phone: 505-889-4543  E-mail: john@aquarius.unm.edu -or- jkprent@cs.sandia.gov