Gokhan said:

"I need Gauss-Jackson integration algorithm. Although I searched so much on the internet- I could not find any resource that explicitly explains the algorithm. Can you advise me a reference- a book may be, or a web site- either way it is fine..."

I'm reading this from my home machine (my comments below are based upon my faulty memory). My reference material is at work. I'll try to find a reference soon. What type of trajectory are you trying to integrate? If it's straight celestial mechanics then you should consider using a perturbation method. If you're modeling a spacecraft trajectory that's out of the atmosphere then you'll probably need to use Gauss-Jackson. If you want super accuracy then you'll need to go to 8th order. If it's in the atmosphere then you should use a Fehlberg Runge-Kutta.

Gauss-Jackson is a predictor-corrector method that is not self starting. You need to use a Runge-Kutta of similar order to load up a continuation matrix. The Runge-Kutta is like the electic starter motor on an automobile engine which is the Gauss-Jackson predictor. The logic gets really tricky if you want to use a variable step size control algorithm (this is done using different solution orders and then measuring the difference between the results). Gauss-Jackson is mainly useful for calculating orbital mechanics. It integrates directly from the second derivative, e.g. Y''=F(Y). I believe(?) Herrick wrote about Gauss-Jackson. Good old Carl Friedrich Gauss originally invented it, so you know it's got to be good.

Gary