求解大型稀疏线性方程组是许多科学和工程计算中最重要的问题之一,Krylov子空间方法是求解这类线性方程组的一个研究热点.本文介绍了Krylov子空间方法及其分类,例如正交投影方法(或Ritz-Galerkin方法),正交化方法(或极小残差方法),双正交化方法(或Petrov-Galerkin方法),解法方程组的CGNE和CGNR方法等,指出了这些方法在算法设计方面国内外研究现状和存在问题,着重考虑稀疏矩阵向量乘积与内积计算方法的并行处理问题;讨论了预条件与并行预条件技术,残差磨光技术及其并行实现,数据的合理分布问题,内积瓶颈问题等方面研究的发展趋势,希望有更多学者了解和研究这些方法.

Solving a large sparse linear system of equations is one of the most important problems in scientific and engineering computations. The Krylov subspace methods are widely used in this respect. This paper first reviews the Krylov subspace methods and their various types, such as, the orthogonal projection method (Ritz-Galerkin method), the orthogonalization method (or the minimal residual method), the bi-orthogonalization method (Petrov-Galerkin method), and the CGNE and CGNR methods for normal systems. The advantages and shortcomings of these methods are analyzed. Especially, we focus on the parallel computation of the sparse matrix-vector multiplication and the inner product. Then, this paper discusses the development of the preconditioning and the parallel preconditioning technique, the residual smoothing technology with its parallel implementation, the reasonable distribution of data, the bottleneck problem of the inner product.