Proof of convergence

[mfkasim1]

• [x] If the solution exists, can we proof that the convergence always exists?
• [ ] If the convergence does not always exist, what can we do to make the convergence always exist?

[mfkasim1]

If the solution exists, can we proof that the convergence always exists?

The perturbation theory only leads to fixed-point iteration of y = g[y] = L0_inv[f] - L0_inv_L[sigma(y)]. mode=1 only leads to the naive fixed-point iteration which only works if the absolute eigenvalue of the Jacobian of g[y] less than 1. mode=2 leads to linear mixing, which is a bit better, but not so much. With linear mixing, you can only converge if the largest eigenvalue of the Jacobian satisfies 1 < largest_eival < -(1 + eps) / (1 - eps). So, the convergence does not always exists. Unless maybe if we do some nonlinear preconditioning on the equation.

[mfkasim1]

So many things happened after this and this issue becomes unrelated.