We propose an inexact coordinate descent method to solve a consistent equation of the type Ax = b where A is a real symmetric and positive semidefinite matrix. The admissible range of inexactitude allows to reduce every multiplication present in the computation to a scaling by a signed power of 2 for the sake of hardware simplification. Although the resulting descent is nonideal, the numerical experiments show that its rate of convergence remains close to that of exact coordinate descent, whether the coordinate selection is cyclic or random. Meanwhile, our algorithm outperforms both methods with an additional low complexity “weak greedy” selection of the coordinates, also based on scaling by signed powers of 2. Our method is based on an advanced use of varying relaxation coefficients in the Gauss–Seidel iteration, with special theoretical considerations when A is singular.

中文翻译：

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用于求解半定线性系统的粗算术坐标下降

我们提出了一种不精确坐标下降法来求解类型为Ax  =  b的一致方程，其中A是实对称半正定矩阵。为了硬件简化，允许的不精确范围允许将计算中存在的每个乘法减少到按 2 的有符号幂进行缩放。尽管由此产生的下降是非理想的，但数值实验表明，无论坐标选择是循环的还是随机的，其收敛速度仍然接近精确坐标下降的收敛速度。同时，我们的算法通过额外的低复杂度“弱贪婪”坐标选择优于这两种方法，也是基于 2 的有符号幂的缩放。我们的方法基于在高斯-赛德尔迭代中对不同松弛系数的高级使用，当A是单数时，有特殊的理论考虑。