both Barzilai-Borwein algs have been written and tested.
Matthieu informed me he intends to rename some variables, especially containing word "step" because it yields a big ambiguity (alpha_step or x_step, and some more). previously direction also had "step" in variable name.
However, I have some problems with default sigma1, sigma2, alpha1, alpha2, rho, sigma values (from non-monotone Barzilai-Borwein alg). My initial guesses were too bad, as I noticed. Currently it works so-so - but maybe only for my tests.