GPBi-CG: Generalized Product-type Methods Based on Bi-CG for Solving Nonsymmetric Linear Systems

Abstract

Recently Bi-CGSTAB as a variant of Bi-CG has been proposed for solving nonsymmetric linear systems, and its attractive convergence behavior has been confirmed in many numerical experiments. Bi-CGSTAB can be characterized by its residual polynomial which consists of the product of the residual polynomial of Bi-CG with other polynomials generated from two-term recurrence relations. In this paper, we propose a unified way to generalize a class of product-type methods whose residual polynomials can be factored by the residual polynomial of Bi-CG and other polynomials with standard three-term recurrence relations. Such product-type methods which are based on Bi-CG can be regarded as generalizations of Bi-CGSTAB. From the unified way, the well-known variants of the product-type methods, like CGS, Bi-CGSTAB, Bi-CGSTAB2, are reacquired again.

The discrete dipole approximation: An overview and recent developments

“…At first, they were used to accelerate computations [1], but they also allowed larger numbers of dipoles to be simulated [6,108], since storage of the entire matrix is prohibitive for direct methods. The most widely used iterative methods in the DDA are Krylov-space methods, such as [107] conjugate gradient (CG), CG applied to the Normalized equation with minimization of Residual norm (CGNR), Bi-CG, Bi-CG stabilized (Bi-CGSTAB), CG squared (CGS), generalized minimal residual (GMRES), quasi-minimal residual (QMR), transpose free QMR (TFQMR), and generalized product-type methods based on Bi-CG (GPBi-CG) [109].…”

BiCGCR2: A new extension of conjugate residual method for solving non-Hermitian linear systems

“…Therefore it can be used as a basic iterative procedure for the development of other non-optimal Krylov subspace methods, similarly to the BiCG, BiCR and BiCOR algorithm that have motivated the development of BiCGSTAB(ℓ) (GPBiCG) [23,25], BiCRSTAB(ℓ) (GPBi-CR) [16,32] and BiCORSTAB2 (GPBiCOR) [34,35,53]. In future work, we plan to construct hybrid variants of BiCGCR2, for which r n := H n (A)r BiCGCR2 n where H n is a suitable matrix polynomial of degree n, along the same lines of the derivations of hybrid BiCG, hybrid BiCR or hybrid BiCOR.…”

Parallel Computing of Fluid-Structure Coupled Analysis Using Supg/PSPG and Enriched Free Mesh Method

“…Agter discretizing this formulation in the time direction, a system of linear equations is deribed with the unknown variable U n+1 , V n+1 and P n+1 in the two-dimensional case. As the matrices in the above equations are asymmetric, we adopt the Generalized Product type method based on the Bi-CG)GPBi-CG) [22,23] method or the General Minimal Residual (GMRES(m)) method.…”