<p>The harmonic balance method with alternating frequency-time scheme (HBM-AFT) implements the Fourier Transform to obtain harmonic coefficients of frictional hysteretic forces from their corresponding time-domain solutions. However, expensive computation of these hysteretic forces and their gradients limits the efficiency of HBM-AFT. This paper introduces a customized state-space algorithm and integrates it into the HBM-AFT, enabling fast friction force calculation and efficient analysis of periodic solutions. To this end, a zero-one representation is addressed to describe the stick-slip behaviors of friction elements, resulting in uncoupling piece-wise state-space formulations. Subsequently, the stick-slip transition points tracking strategy is developed and combined with the precise time step integration to solve this formulation step by step. These procedures formulate the customized state-space algorithm that efficiently computes the periodic hysteretic forces within the quasi-static framework provided by the HBM-AFT. Furthermore, the gradient of hysteretic forces with respect to the harmonic coefficients is evaluated using an equivalent linearization method. This method straightforwardly derives the generalized damping and stiffness from frictional hysteretic curves, enabling the improvement of the harmonic solutions convergence in the Newton-Raphson iteration. Structures with Jenkins friction and Iwan bolted joints are studied to verify the accuracy and efficiency of the proposed method.</p>