Welcome to Gaia! ::

Making this thread to discuss various numerical methods and post algorithm outlines or pseudocode. I just started needing more sophisticated/complex/efficient numerical methods to do research, so I feel pretty new to the subject. There should be quite a few of you out there with some tips and tricks. I'll start off with a few questions:

1) People's favorite methods book(s)? Mine so far has been Wikipedia, which may explain why I mess up algorithms so often XD

2) For those who know the conjugate gradient method on normal equations, is there a nice general preconditioner like the Jacobi preconditioner? For a finite difference PDEs that can't produce symmetric, positive definite or diagonally-dominant matrices it seems like this would be a good method but the condition number's too high.

3) Geometric integration methods seem like a kewl idea and relatively straightforward to implement for problems that can be phrased in Hamiltonian form. What if you have a more complex system with only a small number of conserved quantities? What would you do?

4) Finally, suppose you have a system that can be phrased in Lagrangian terms, i.e. find the function which maximizes a functional, but for whatever reason the associated differential equation is too complicated, messy, etc. Multigrid methods would work well for approximating the function, since you could use previous solutions later, but what if you want to use an adaptive mesh? How would you go about doing it?

Again, feel free to raise questions, discuss, post code, etc.