Components based on shape-memory alloys are often subjected to several loading cycles that result in substantial alteration of material behavior. In such a framework, accurate models as well as robust and efficient numerical approaches become essential to allow for the simulation of complex devices. The present paper focuses on the numerical simulation of quasi-static problems involving shape memory alloy (SMA) structures or components subjected to multiple loading-unloading cycles. A novel state-update procedure for a three-dimensional phenomenological model able to describe the saturation of permanent inelasticity, including degradation effects, is here proposed. The algorithm, being of the predictor-corrector type and relying on an incremental energy minimization approach, is based on elastic checks, closed-form solutions of polynomial equations, and nonlinear scalar equations solved through a combination of Newton-Raphson and bisection methods. This allows for an easy implementation of model equations and to avoid the use of regularization parameters for the treatment of non-smooth functions. Numerical results assess the good performances of the proposed approach in predicting both pseudoelastic and shape-memory material behavior under cyclic loading as well as algorithm robustness.