Continued Roots, Power Transform and Critical Properties

We consider the problem of calculation critical amplitudes at infinity by means self-similar continued root approximants. Region applicability approximants is extended from determinate (convergent) with well-defined conditions studied before Gluzman and Yukalov (Phys. Lett. A 377 2012, 124), to indeterminate (divergent) my power transformation. Most challenging for roots problems calculating amplitudes, can be successfully attacked performing proper transformation found optimization imposed on parameters transform. The were derived systematically applying algebraic renormalization each every level interactions their strength increasing, while follows fundamental symmetry principle functional self-similarity, realized constructively in space approximations. Our approach solution replace it problem, but some unknown control parameter b place known index β. From way making convergent. β hidden under carpet replaced b. idea applied various, mostly quantum-mechanical problems. In particular, method allows us solve Bose-Einstein condensation temperature good accuracy.