Lanczos‐type product methods (LTPMs), in which the residuals are defined by the product of stabilizing polynomials and the Bi‐CG residuals, are effective iterative solvers for large sparse nonsymmetric linear systems. Bi‐CGstab(L) and GPBi‐CG are popular LTPMs and can be viewed as two different generalizations of other typical methods, such as CGS, Bi‐CGSTAB, and Bi‐CGStab2. Bi‐CGstab(L) uses stabilizing polynomials of degree L, while GPBi‐CG uses polynomials given by a three‐term recurrence (or equivalently, a coupled two‐term recurrence) modeled after the Lanczos residual polynomials. Therefore, Bi‐CGstab(L) and GPBi‐CG have different aspects of generalization as a framework of LTPMs. In the present paper, we propose novel stabilizing polynomials, which combine the above two types of polynomials. The resulting method is referred to as GPBi‐CGstab(L). Numerical experiments demonstrate that our presented method is more effective than conventional LTPMs.

中文翻译：

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GPBi‐CGstab（L）：一种将Bi‐CGstab（L）和GPBi‐CG结合起来的Lanczos型产品方法

Lanczos型乘积方法（LTPM）的残差由稳定多项式与Bi-CG残差的乘积定义，是适用于大型稀疏非对称线性系统的有效迭代求解器。Bi‐CGstab（L）和GPBi‐CG是流行的LTPM，可以看作是其他典型方法的两种不同概括，例如CGS，Bi‐CGSTAB和Bi‐CGStab2。Bi‐CGstab（L）使用度为L的稳定多项式，而GPBi‐CG使用以Lanczos残差多项式为模型的三项递归（或等效的耦合二项递归）给出的多项式。因此，Bi-CGstab（L）和GPBi‐CG具有作为LTPM框架的泛化的不同方面。在本文中，我们提出了新颖的稳定多项式，其结合了以上两种类型的多项式。生成的方法称为GPBi-CGstab（L）。数值实验表明，我们提出的方法比传统的LTPM更有效。