# IDR(s): A Family of Simple and Fast Algorithms for Solving Large Nonsymmetric Systems of Linear Equations

@article{Sonneveld2008IDRsAF,
title={IDR(s): A Family of Simple and Fast Algorithms for Solving Large Nonsymmetric Systems of Linear Equations},
author={Peter Sonneveld and Martin B. van Gijzen},
journal={SIAM J. Sci. Comput.},
year={2008},
volume={31},
pages={1035-1062}
}
• Published 1 November 2008
• Computer Science
• SIAM J. Sci. Comput.
We present IDR($s$), a new family of efficient, short-recurrence methods for large nonsymmetric systems of linear equations. The new methods are based on the induced dimension reduction (IDR) method proposed by Sonneveld in 1980. IDR($s$) generates residuals that are forced to be in a sequence of nested subspaces. Although IDR($s$) behaves like an iterative method, in exact arithmetic it computes the true solution using at most $N + N/s$ matrix-vector products, with $N$ the problem size and $s… 266 Citations ## Figures and Tables from this paper • Computer Science, Mathematics TOMS • 2011 This article derives a new IDR(s) variant, that imposes (one-sided) biorthogonalization conditions on the iteration vectors, and shows that the new variant is more stable and more accurate than the originals, and that it outperforms other state-of-the-art techniques for realistic test problems. • Computer Science SIAM J. Sci. Comput. • 2010 Through numerical experiments it is shown that IDRstab can outperform both IDR($s$) and BiCGstab($\ell$), and the relation between hybrid Bi-CG methods and IDR and the new concept of the Sonneveld subspace as a common framework is introduced. • Computer Science • 2010 This paper reformulated the efficient and stable IDR(s) algorithm in such a way that it has a single global synchronisation point per iteration step and a methodology is presented for a–priori estimation of the optimal value of s using only problem and machine–based parameters. • Mathematics • 2012 IDR (Induced Dimension Reduction) is a family of efficient iterative methods for the numerical solution of large non-symmetric systems Ax = b of linear equations. Examples of IDR methods are • Computer Science Numer. Linear Algebra Appl. • 2011 This paper reformulated a recently proposed IDR (s) algorithm that is highly efficient and stable is reformulated in such a way that it has a single global synchronization point per iteration step. ## References SHOWING 1-10 OF 24 REFERENCES The author presents for real nonsymmetric matrices a method BICGSTAB2 in which the second factor may have complex conjugate zeros, and versions suitable for complex matrices are given for both methods. This paper proposes a unified way to generalize a class of product-type methods whose residual polynomials can be factored by the residualPolynomial of Bi-CG and other polynmials with standard three-term recurrence relations. • Mathematics • 1984 We characterize the class$CG(s)$of matrices A for which the linear system$A{\bf x} = {\bf b}\$ can be solved by an s-term conjugate gradient method. We show that, except for a few anomalies, the
• Mathematics
Overton, who showed us how the ideal Arnoldi and GMRES problems relate to more general problems of minimization of singular values of functions of matrices 17]. gmres and arnoldi as matrix
In an earlier publication [14] a method was described which generated the eigenvalues and eigenvectors of a matrix by a successive algorithm based on minimizations by least squares. The advantage of
• Computer Science
SIAM J. Sci. Comput.
• 1999
A variant of the popular BiCGSTAB method for solving nonsymmetric linear systems that produces a residual polynomial which is of lower degree after the same number of steps, but which also requires fewer matrix-vector products to generate, on average requiring only 1+1/k matvecs per step.
• Computer Science
• 1991
A novel BCG-like approach, the quasi-minimal residual (QMR) method, which overcomes the problems of BCG is presented and how BCG iterates can be recovered stably from the QMR process is shown.
Numerical experiments indicate that the new variant of Bi-CG, named Bi- CGSTAB, is often much more efficient than CG-S, so that in some cases rounding errors can even result in severe cancellation effects in the solution.
• Mathematics
Numerical Algorithms
• 2005
This paper will propose a strategy for a more stable determination of the Bi-CG iteration coefficients and by experiments it will show that this indeed may lead to faster convergence.
A Lanczos-type method is presented for nonsymmetric sparse linear systems as arising from discretisations of elliptic partial differential equations. The method is based on a polynomial variant of