Low-precision computing is essential for efficiently utilizing memory bandwidth and computing cores. While many mixed-precision algorithms have been developed for iterative sparse linear solvers, effectively leveraging half-precision (fp16) arithmetic remains challenging. This study introduces a novel nested Krylov approach that integrates the FGMRES and Richardson methods in a deeply nested structure, progressively reducing precision from double-precision to fp16 toward the innermost solver. To avoid meaningless computations beyond precision limits, the low-precision inner solvers perform only a few iterations per invocation, while the nested structure ensures their frequent execution. Numerical experiments show that incorporating fp16 into the approach directly enhances solver performance without compromising convergence, achieving speedups of up to 1.65 × and 2.42 × over double-precision and double-single mixed-precision implementations, respectively. Furthermore, the proposed method outperforms conventional mixed-precision Krylov solvers, CG, BiCGStab, and restarted FGMRES, by factors of up to 2.47, 2.74, and 69.10, respectively.

We consider solving a sparse linear system using integer (fixed-point) arithmetic. Integer arithmetic has attracted attention in scientific computing because of its high computational efficiency. Furthermore, considering the current circumstances of hardware development, integer arithmetic is expected to become increasingly important. Nevertheless, integer arithmetic has not been widely used for solving linear systems because it lacks robustness against overflow and underflow, making it hard to solve practical problems. Thus, we propose a new integer-based implementation framework for the Flexible GMRES (FGMRES) method, which enables integer-based solvers to solve linear systems with the same accuracy as conventional floating-point solvers. In addition, we propose an integer-only algebraic multigrid preconditioner. Combining it with the integer-based FGMRES framework, we develop an integer-based solver. Numerical experiments on CPUs showed that the developed integer-based solver has a comparable convergence rate to floating-point solvers. We also found the test cases where the integer-based solver runs faster than the floating-point solvers.

In this paper, a novel teaching material for the subject in Electric Power Engineering is proposed. The content of the subject is a non-practical, because the really necessary equipment cannot be installed on campus. Therefore, while taking the subject, students can neither feel the atmosphere nor know the danger of it. As a result of various trial and error, the authors have developed innovative teaching materials for introducing the subject. This teaching material employs a board game simulation format, so students can learn with interest. By incorporating digital contents into the tutorial, the simulation can be more facilitated. In addition, by linking with the active text system (A-txt) that is a digital teaching material using AR technology, the additional information can be browsed by holding their own smart terminal over the game sheets. In this time, it was confirmed that by applying the gaming simulation to the KOSEN students. They have been able to understand the learning goals represented by the difficulty of achieving 3E and the different characteristics of each country’s situation. Consequently, it was shown that students could study global issues in environmental and energy problems beyond the learning targets using the proposed material.