Abstract

Moment methods in Krylov subspaces for solving symmetric systems of linear algebraic equations (SLAEs) are considered. A family of iterative algorithms is proposed based on generalized Lanczos orthogonalization with an initial vector \({{v}^{0}}\) chosen regardless of the initial residual. By applying this approach, a series of SLAEs with the same matrix, but with different right-hand sides can be solved using a single set of basis vectors. Additionally, it is possible to implement generalized moment methods that reduce to block Krylov algorithms using a set of linearly independent guess vectors \(v_{1}^{0},...,v_{m}^{0}.\) The performance of algorithm implementations is improved by reducing the number of matrix multiplications and applying efficient parallelization of vector operations. It is shown that the applicability of moment methods can be extended using preconditioning to various classes of algebraic systems: indefinite, incompatible, asymmetric, and complex, including non-Hermitian ones.

中文翻译：

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关于Krylov子空间的矩方法

摘要

考虑了Krylov子空间中用于求解线性代数方程（SLAE）对称系统的矩方法。提出了一种基于广义Lanczos正交化的迭代算法族，该迭代算法选择了与初始残差无关的初始向量\（{{v} ^ {0}} \）。通过应用此方法，可以使用一组基本向量来求解具有相同矩阵但右侧不同的一系列SLAE。另外，可以使用一组线性独立的猜测矢量\（v_ {1} ^ {0}，...，v_ {m} ^ {0}。\}来实现广义矩方法，从而减少以阻塞Krylov算法。通过减少矩阵乘法的数量并应用矢量运算的有效并行化，可以提高算法实现的性能。结果表明，矩方法的适用性可以通过预处理扩展到各种类型的代数系统：不确定，不兼容，不对称和复杂，包括非Hermitian系统。