We propose a general application programming interface called OpenATLib for auto-tuning (AT). OpenATLib is carefully designed to establish the reusability of AT functions for sparse iterative solvers. Using APIs of OpenATLib, we develop a fully auto-tuned sparse iterative solver called Xabclib. Xabclib has several novel runtime AT functions. We also develop a numerical computation policy that can optimize memory space and computational accuracy. Using the above functions and policies, we obtain the following important findings: (1) an average memory space is reduced to 1/45 under lower memory policies, and (2) fault convergence, which the conventional solvers judges to be converged but actually not converged in the sense of the before-preconditioned matrix, is avoided under higher accuracy policies. The results imply policy-based runtime AT plays significant role in sparse iterative matrix computations.