This paper proposes a verification method for sparse linear systems $Ax=b$ with general and nonsingular coefficients. A verification method produces the error bound for a given approximate solution. Conventional methods use one of two approaches. One approach is to verify the computed solution of the normal equation $A^TAx=A^Tb$ by exploiting symmetric and positive definiteness; however, the condition number of $A^TA$ is the square of that for $A$. The other approach uses an approximate inverse matrix of the coefficient; however, the approximate inverse may be dense even if $A$ is sparse. Here, we propose a method for the verification of solutions of sparse linear systems based on $LDL^T$ decomposition. The proposed method can reduce the fill-in and is applicable to many problems. Moreover, an efficient iterative refinement method is proposed for obtaining accurate solutions.

翻译：本文提出了一种针对具有一般非奇异系数的稀疏线性方程组$Ax=b$的验证方法。验证方法旨在为给定的近似解提供误差界。传统方法采用两种途径之一：一种途径是通过利用对称正定性来验证正规方程$A^TAx=A^Tb$的计算解；然而，$A^TA$的条件数是$A$条件数的平方。另一种途径使用系数矩阵的近似逆矩阵；但即使$A$是稀疏的，其近似逆矩阵也可能是稠密的。本文提出了一种基于$LDL^T$分解的稀疏线性方程组解验证方法。所提方法能够减少填充量，并适用于多种问题。此外，本文还提出了一种高效的迭代精化方法以获得精确解。