We introduce an iterative solver named MINARES for symmetric linear systems $Ax \approx b$, where $A$ is possibly singular. MINARES is based on the symmetric Lanczos process, like MINRES and MINRES-QLP, but it minimizes $\|Ar_k\|$ in each Krylov subspace rather than $\|r_k\|$, where $r_k$ is the current residual vector. When $A$ is symmetric, MINARES minimizes the same quantity $\|Ar_k\|$ as LSMR, but in more relevant Krylov subspaces, and it requires only one matrix-vector product $Av$ per iteration, whereas LSMR would need two. Our numerical experiments with MINRES-QLP and LSMR show that MINARES is a pertinent alternative on consistent symmetric systems and the most suitable Krylov method for inconsistent symmetric systems. We derive properties of MINARES from an equivalent solver named CAR that is to MINARES as CR is to MINRES, is not based on the Lanczos process, and minimizes $\|Ar_k\|$ in the same Krylov subspace as MINARES. We establish that MINARES and CAR generate monotonic $\|x_k - x_{\star}\|$, $\|x_k - x_{\star}\|_A$ and $\|r_k\|$ when $A$ is positive definite.

翻译：本文介绍了一种名为MINARES的迭代求解器，用于求解对称线性系统$Ax \approx b$，其中$A$可能为奇异矩阵。MINARES基于对称Lanczos过程（类似于MINRES和MINRES-QLP），但它在每个Krylov子空间中最小化的是$\|Ar_k\|$而非$\|r_k\|$，其中$r_k$为当前残差向量。当$A$对称时，MINARES与LSMR最小化相同的量$\|Ar_k\|$，但在更相关的Krylov子空间中进行，且每次迭代仅需一次矩阵-向量乘积$Av$，而LSMR需要两次。我们与MINRES-QLP及LSMR的数值实验表明：在一致对称系统中，MINARES是一种有效的替代方案；在非一致对称系统中，它是最适宜的Krylov方法。我们从等价求解器CAR推导出MINARES的性质——CAR之于MINARES相当于CR之于MINRES，它不基于Lanczos过程，但在与MINARES相同的Krylov子空间中最小化$\|Ar_k\|$。我们证明：当$A$正定时，MINARES和CAR生成的$\|x_k - x_{\star}\|$、$\|x_k - x_{\star}\|_A$及$\|r_k\|$均具有单调性。