ODE-based steadystate solving method

[ytdHuang]

The current existing steadystate solving methods doesn't depend on the initial condition (state), but there are some cases which leads to different stationary states because of the initial states.

By the multiple-dispatch feature of Julia, we can have a new steadystate method with a given initial state.
The stationary state can be obtained based on DiffEqCallbacks.jl and the solvers in OrdinaryDiffEq.jl.

For the terminate condition of finding the stationary state, it would be better to use:

either the following condition is true
$$\lVert\frac{\partial |\rho\rangle\rangle}{\partial t}\rVert \leq \textrm{reltol} \times\lVert\frac{\partial |\rho\rangle\rangle}{\partial t}+|\rho\rangle\rangle\rVert$$

or

$$\lVert\frac{\partial |\rho\rangle\rangle}{\partial t}\rVert \leq \textrm{abstol}$$

[aarontrowbridge]

I have begun working on this issue as well. Do you have a degenerate example system in mind to test on?

[ytdHuang]

I have begun working on this issue as well. Do you have a degenerate example system in mind to test on?

I don't have a concrete example now.

By utilizing the ODE method, the result is guaranteed to be unique (with the given Liouvillian and initial state).

However, the SteadyStateDirectSolver based on LinearSolve.jl doesn't guarantee to have a unique solution.

The result obtained from the ODE solver should be the same as the one obtained from mesolve (with long enough time evolution).

Besides, I think someone else made a PR #159 ?

[ilkclord]

This is a comment to allow the Unitary Hack bots to find the PR above ( #159 )