A sparse partial update (SPU) algorithm and its improved version improved SPU (ISPU) algorithm, are proposed in this paper for sparse system identification. The SPU first categorizes its filter coefficients into active and inactive coefficients. Then all the active coefficients are included in each adaptation, while only a small portion of the inactive coefficients is periodically chosen to be included in the adaptation. The SPU emphasizes convergence of the active coefficients by updating them at every adaptation, while the periodical adaptation of the inactive coefficients ensures its robustness and tracking capability. By eliminating most of the inactive coefficients from adaptation, the SPU significantly reduces the number of adapting coefficients in each adaptation. The decline in the number of adapting coefficients eventually leads to improvement in both computation and convergence speed. To avoid performance degradation in the case of identifying a dispersive system, an ISPU is further proposed by making modifications to the SPU. The simulation results demonstrate that the ISPU not only outperforms other sparse adaptive algorithms in identifying a sparse system but also performs at least as well as the NLMS algorithm in identifying a dispersive system.