SIAM Journal on Scientific Computing, Ahead of Print.
For the solution of large sparse unsymmetric systems, Duff et al. [SIAM J. Sci. Comput., 37 (2015), pp. A1248--A1269] proposed an approach based on the block Cimmino iterations [Numer. Math., 35 (1980), pp. 1--12], in which the solution is computed in a single iteration, so we call it a pseudodirect solver. In this approach, matrices are augmented with additional variables and constraints so that a partitioning of the matrix in blocks of rows defines mutually orthogonal subspaces. The augmented system can then be solved efficiently with a sum of projections onto these orthogonal subspaces. The purpose of this manuscript is to extend this method to the minimum norm solution of underdetermined systems and to the solution of least-squares problems. In the latter case, a partitioning of the matrix in blocks of columns rather than rows is used, and the system must be suitably augmented to define mutually orthogonal subspaces again to recover the least-squares solution of the original problem. This article proves the equivalence between the solution of the original and the augmented system. In order to complete the extension to overdetermined systems, we also propose an iterative block conjugate gradient acceleration [SIAM J. Sci. Comput., 16 (1995), pp. 1478--1511] for the solution of least-squares problems. The efficiency of both the iterative and the augmented pseudodirect approaches, as implemented in the ABCD-Solver, is illustrated on large rectangular matrices from the SuiteSparse Matrix Collection. -We have learnt with great sadness that our co-author, long-term collaborator, advisor, and good friend Constantin Popa passed away while we were revising this article. We would like to dedicate this work to his memory.

中文翻译：

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增广块 Cimmino 方法对满秩矩形系统解的扩展

SIAM 科学计算杂志，提前印刷。
对于大型稀疏非对称系统的解决方案，Duff 等人。[SIAM J. Sci. Comput., 37 (2015), pp. A1248--A1269] 提出了一种基于块 Cimmino 迭代的方法 [Numer. Math., 35 (1980), pp. 1--12]，其中的解在单次迭代中计算，因此我们称其为伪直接求解器。在这种方法中，矩阵增加了额外的变量和约束，以便矩阵在行块中的划分定义了相互正交的子空间。然后可以通过对这些正交子空间的投影求和来有效地求解增强系统。本手稿的目的是将此方法扩展到欠定系统的最小范数解和最小二乘问题的解。在后一种情况下，使用列块而不是行块对矩阵进行分区，并且系统必须适当地扩充以再次定义相互正交的子空间以恢复原始问题的最小二乘解。本文证明了原始系统和增强系统的解决方案之间的等价性。为了完成对超定系统的扩展，我们还提出了一种迭代块共轭梯度加速[SIAM J. Sci. Comput., 16 (1995), pp. 1478--1511] 用于解决最小二乘问题。在 ABCD-Solver 中实现的迭代和增强伪直接方法的效率在 SuiteSparse 矩阵集合中的大型矩形矩阵上进行了说明。- 我们非常悲伤地了解到，我们的合著者、长期合作者、顾问和好朋友康斯坦丁·波帕在我们修改这篇文章时去世了。