Abstract
Buttari et al. have proposed the mixed-precision iterative refinement method using the IEEE754 single- and double-precision arithmetic for solving linear systems of equations. Their benchmark tests show that their proposed method can obtain high performance on computation environment on which the performance of single-precision arithmetic can be higher than double-precison arithmetic. However, the application of their method is limited to well-conditioned problems that can be solved by using single-precision arithmetic. Hence, the convergence of ill-conditioned problems may not be possible, which requires double-precision arithmetic. We broaden the scope of applications of the mixed-precision iterative refinement method by using a combination of double-precision arithmetic and multiple-precision arithmetic, and show that the new method has higher performance and yields more precise solutions than the original method. Finally, throught our numerical experiments, we demonstrate that the implicit Runge-Kutta methods with the mixed-precision iterative refinement method can speed up.
