Preconditioning

 

When problems have less than about 10³ unknowns, ``exact'' methods are generally satisfactory. When there are more unknowns, we use iterative methods. Iterative methods are about as fast as exact methods, but they allow us to hope that when we give up exhausted that we are near the best that can be done. With more experience, I learned that we often remain far from the solution we seek. Thus we seek generally applicable problem reformulations that will converge more rapidly. Luckily we find that the helix gives a huge speedup in many problems, in those problems that are densely sampled in model space.