GSoC 2018 - Week 9 - Starting up with Lambert equations

13 Jul 2018

At the start of the week I worked on the leftovers of week 8:

• added log_singularities() that will help in determining logarithmic singularities,
• improved documentation of all helpers as suggested by Amit to maintain consistency

Status of the PR’s:

PR #14736 is ready to be merged.

PR #14792 is being worked on. Major tasks has been completed, just review and refining has to be done.

Apart from this I started working on the way Lambert type equations can be solved through _transolve(). I looked into _tsolve's way of handling such equations. For solving Lambert type equations _tsolve() largely depends on bivariate.py. It takes help of the different utility functions implemented there. Of them two important are _solve_lambert() and bivariate_type(). These two helpers help in getting the equations evaluated.

Equations that can be written in the standard form as: A + B*x + C*log(D + E*x) = 0 has the solutions in terms of Lambert function as:

D/E + C*B*W(t) with (B/C*E)*exp((BD - AE)/CE)

This is what _solve_lambert() determines and accordingly returns the solutions, otherwise returns a NotImplementedError

If _solve_lambert() is unable to handle bivariate_type() is tried. This function first tries to identify the type of composite bivariate and then substitutes Dummy in place of them. For eq: (x + y)**2 - 3 would become _u**2 - 3 where _u is the dummy variable. The idea is that solving the latter equation for u and then equating the solutions to the former equation is equivalent for solving the original one.

While implementing in _transolve this philosophy needs to be applied. As of now I have looked into different tests on how they behave. I will start implementing it next.

Next week’s plan:

• Finishing with the logsolver

• Implementing lambert solver.

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