To address the issue of a sharp decline in convergence speed of the classical Kaczmarz method when solving asymptotically singular linear equations, a robust algorithm with a consistently stable convergence rate is proposed. By regarding the row space of the coefficient matrix as a subspace in the Grassmann manifold, a new concept of asymptotically singular matrices with row rank degeneracy is introduced, and the asymptotic null space and asymptotic dual null space are defined. Based on this, a robust Kaczmarz method is proposed through the subspace correction framework, and the convergence analysis is completed using the Xu-Zikatanov identity. Finally, based on the equivalence between the Kaczmarz method and the coordinate descent method and the relationship between the asymptotic null space and the dual null space, a robust coordinate descent algorithm is proposed, and the equivalence between the robust Kaczmarz method and the robust coordinate descent algorithm is proved. Theoretical analysis shows that the proposed robust Kaczmarz method and robust coordinate descent algorithm have a consistently stable convergence rate when solving asymptotically singular linear equations. Numerical experiments verify the robustness and effectiveness of the algorithms.