A novel approach for Milne's phase-amplitude method

We have uncovered a linear equation for the envelope function---fully equivalent with the original non-linear equation of Milne's---and have implemented a highly accurate and efficient numerical method for computing the envelope and the associated phase. Consequently, we obtain a high precision parametrization of the wavefunction, within a very economical approach. The key ingredients are: (i) straightforward optimization for smoothness, and (ii) Chebyshev polynomials as the workhorse for solving integro/differential equations. The latter also give a built-in interpolation, and allow for developing numerical tools that are robust, accurate, and convenient.