Large and sparse nonsymmetric linear systems of the form Ax = b are essential problems in scientific computing, and Krylov subspace methods for solving the linear systems are recognized as one of the “Top 10 Algorithms having the greatest influence on the development and practice of science and engineering in the 20th century” [1]. For details on Krylov subspace methods including the recent development, see, e.g., [2]. Toward the end of the 20th century, S.-L. Zhang constructed a framework that successfully incorporates the CGS and Bi-CGSTAB methods and subsequently introduced the GPBi-CG method derived from the same framework [3]. While the GPBi-CG method often converges faster than the CGS and Bi-CGSTAB methods, there are still cases where the CGS method performs the best. This observation has motivated us to revisit the framework to find a hybrid algorithm, combining the GPBi-CG method and the CGS method. In this talk, we introduce the hybrid algorithm, showing its efficiency in various numerical experiments, and discuss the reasons behind its effectiveness. REFERENCES [1] J. Dongarra and F. Sullivan, Guest editors introduction: The top 10 algorithms, Comput. Sci. Eng. 2, pp. 22–23, 2000. [2] T. Sogabe, Krylov Subspace Methods for Linear Systems: Principles of Algorithms, Springer, 2023. [3] S.-L. Zhang, GPBi-CG: generalized product-type methods based on Bi-CG for solving nonsymmetric linear systems, SIAM J. Sci. Comput., Vol. 18, pp. 537–551, 1997.