# 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} }

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

### A block IDR(s) method for nonsymmetric linear systems with multiple right-hand sides

- Mathematics, Computer ScienceJ. Comput. Appl. Math.
- 2011

### Algorithm 913: An elegant IDR(s) variant that efficiently exploits biorthogonality properties

- Computer Science, MathematicsTOMS
- 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.

### Exploiting BiCGstab(ℓ) Strategies to Induce Dimension Reduction

- Computer ScienceSIAM 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.

### IDR(s) for solving shifted nonsymmetric linear systems

- Computer ScienceJ. Comput. Appl. Math.
- 2015

### Fast solution of nonsymmetric linear systems on Grid computers using parallel variants of IDR(s)

- 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.

### IDR AS A DEFLATION METHOD

- 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…

### Minimizing synchronization in IDR (s)

- Computer ScienceNumer. 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.

### A variant of IDRstab with reliable update strategies for solving sparse linear systems

- Computer ScienceJ. Comput. Appl. Math.
- 2014

### Induced Dimension Reduction Method for Solving Linear Matrix Equations

- Mathematics, Computer ScienceICCS
- 2016

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