CGS, a fast Lanczos-type solver for nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
SIAM Journal on Scientific and Statistical Computing
Variants of BICGSTAB for matrices with complex spectrum
SIAM Journal on Scientific Computing
Generalized conjugate gradient squared
Journal of Computational and Applied Mathematics
GPBi-CG: Generalized Product-type Methods Based on Bi-CG for Solving Nonsymmetric Linear Systems
SIAM Journal on Scientific Computing
QMR Smoothing for Lanczos-Type Product Methods Based on Three-Term Rrecurrences
SIAM Journal on Scientific Computing
ML(k)BiCGSTAB: A BiCGSTAB Variant Based on Multiple Lanczos Starting Vectors
SIAM Journal on Scientific Computing
Look-Ahead Procedures for Lanczos-Type Product Methods Based on Three-Term Lanczos Recurrences
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Scientific Computing
Bi-CGSTAB as an induced dimension reduction method
Applied Numerical Mathematics
A variant of the IDR(s) method with the quasi-minimal residual strategy
Journal of Computational and Applied Mathematics
A block GCROT(m,k) method for linear systems with multiple right-hand sides
Journal of Computational and Applied Mathematics
A variant of IDRstab with reliable update strategies for solving sparse linear systems
Journal of Computational and Applied Mathematics
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IDR($s$) [P. Sonneveld and M. B. van Gijzen, SIAM J. Sci. Comput., 31 (2008), pp. 1035-1062] and BiCGstab($\ell$) [G. L. G. Sleijpen and D. R. Fokkema, Electron. Trans. Numer. Anal., 1 (1993), pp. 11-32] are two of the most efficient short-recurrence iterative methods for solving large nonsymmetric linear systems of equations. Which of the two is best depends on the specific problem class. In this paper we describe IDRstab, a new method that combines the strengths of IDR($s$) and BiCGstab($\ell$). To derive IDRstab we extend the results that we reported on in [G. L. G. Sleijpen, P. Sonneveld, and M. B. van Gijzen, Appl. Numer. Math., (2009), DOI: 10.1016/j.apnum.2009.07.001], where we considered Bi-CGSTAB as an induced dimension reduction (IDR) method. We will analyze the relation between hybrid Bi-CG methods and IDR and introduce the new concept of the Sonneveld subspace as a common framework. Through numerical experiments we will show that IDRstab can outperform both IDR($s$) and BiCGstab($\ell$).