In this paper, we develop a (preconditioned) GMRES solver based on integer arithmetic, and introduce an iterative refinement framework for the solver. We describe the data format for the coefficient matrix and vectors for the solver that is based on integer or fixed-point numbers. To avoid overflow in calculations, we introduce initial scaling and logical shifts (adjustments) of operands in arithmetic operations. We present the approach for operand shifts, considering the characteristics of the GMRES algorithm. Numerical tests demonstrate that the integer arithmetic-based solver with iterative refinement has comparable solver performance in terms of convergence to the standard solver based on floating-point arithmetic. Moreover, we show that preconditioning is important, not only for improving convergence but also reducing the risk of overflow.

中文翻译：

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使用 GMRES 方法和迭代细化的基于整数算术的稀疏线性求解器

在本文中，我们开发了一个基于整数算法的（预处理的）GMRES 求解器，并为求解器引入了迭代优化框架。我们描述了基于整数或定点数的求解器的系数矩阵和向量的数据格式。为了避免计算溢出，我们在算术运算中引入了操作数的初始缩放和逻辑移位（调整）。考虑到 GMRES 算法的特性，我们提出了操作数移位的方法。数值测试表明，具有迭代细化的基于整数算法的求解器在收敛性方面与基于浮点算法的标准求解器具有可比的求解器性能。此外，我们表明预处理很重要，不仅可以提高收敛性，还可以降低溢出的风险。