I have implemented a Quasi Newton method for my problem, where I use the Hessian matrix approximation based approach. Hence, there is a linear system solve in every iteration. I solve the linear system using Conjugate Gradients and use the compact representation given in Representations of quasi-Newton matrices and their use in limited memory methods to get matrix vector products. I am noticing that the linear system is solved in very less number of iterations, 20 to 30 usually. This happens for even very large problems (one million variables). Is this observed in general when working with Quasi Newton? I can think of one possible reason for this fast convergence. Since, I ensure the curvature condition on the Quasi Newton vector pair, the Hessian approximation is guaranteed to be strictly positive definite and the Quadratic form in Conjugate Gradients has a "nice" shape. I am not sure whether this sufficiently explains what is happening.
Edit: Based on the accepted answer below, I have this question. The situation in the Quasi Newton approach, where the linear system can be provably solved in very less number of iterations is very interesting. Are there many works where this has been favorably exploited?