This paper proposes a verification method for sparse linear systems Ax=b with general and nonsingular coefficient matrices. A verification method produces the error bound for a given approximate solution. Practical methods use one of two approaches. One approach is to verify the computed solution of the normal equation ATAx=ATb by exploiting symmetric and positive definiteness; however, the condition number of ATA is the square of that for A. The other approach applies an approximate inverse of A; however, the approximate inverse of A may be dense even if A is sparse. Additionally, several other methods have been proposed; however, they are considered impractical due to various issues. Here, this paper provides a computing method for verified error bounds using the previous verification method and the latest equilibration. The proposed method can reduce the fill-in and is applicable to many problems. Moreover, we will show the efficiency of an iterative refinement method to obtain accurate solutions.

中文翻译：

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具有一般系数的稀疏线性方程组的解验证方法


本文提出了一种具有一般和非奇异系数矩阵的稀疏线性系统 Ax=b 的验证方法。验证方法会生成给定近似解的误差边界。实用方法使用以下两种方法之一。一种方法是通过利用对称和正定性来验证正态方程 ATAx=ATb 的计算解;但是，ATA 的条件编号是 A 的条件编号的平方。另一种方法应用 A 的近似逆函数;但是，即使 A 稀疏，A 的近似逆函数也可能是密集的。此外，还提出了其他几种方法;但是，由于各种问题，它们被认为是不切实际的。在这里，本文提供了一种使用前面的验证方法和最新的均衡来计算验证误差边界的方法。所提方法可以减少填入，适用于许多问题。此外，我们将展示迭代细化方法获得准确解的效率。