The Induced Dimension Reduction method 

IDR(s) is a robust and efficient short recurrence Krylov subspace method for solving large nonsymmetric systems of linear equations. On this page you can find reports and papers that describe IDR(s), MATLAB, Python, and FORTRAN implementations for IDR(s), and examples of how to use the codes.

The software on this page is distributed under the MIT licence:

Copyright (c) 2008 Martin van Gijzen and Peter Sonneveld

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

MATLAB code:




Python code:

Idrs.py: this file contains a Python implementation of IDR(s). It is a direct translation of the matlab implementation described above, with the same functionalities. It has been written and made available by Reinaldo Astudillo (Delft University of Technology).


Fortran code:


Reports and papers:

  1. IDR(s) is described in: Peter Sonneveld and Martin B. van Gijzen, IDR(s): a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations. SIAM J. Sci. Comput. Vol. 31, No. 2, pp. 1035-1062, 2008 (copyright SIAM)

  2. The original IDR(s) report is: Peter Sonneveld and Martin B. van Gijzen, IDR(s): a family of simple and fast algorithms for solving large nonsymmetric linear systems. Delft University of Technology, Reports of the Department of Applied Mathematical Analysis, Report 07-07

  3. The relation of IDR(s) with Bi-CGSTAB, and how to derive generalisations of Bi-CGSTAB using IDR-ideas can be found in: Gerard L.G. Sleijpen, Peter Sonneveld and Martin B. van Gijzen, Bi-CGSTAB as an induced dimension reduction method, Applied Numerical Mathematics. Vol 60, pp. 1100-1114, 2010 (copyright Elsevier)

  4. A very stable and efficient IDR(s) variant (implemented in the MATLAB code idrs.m given above) is described in: Martin B. van Gijzen and Peter Sonneveld, Algorithm 913: An Elegant IDR(s) Variant that Efficiently Exploits Bi-orthogonality Properties. ACM Transactions on Mathematical Software, Vol. 38, No. 1, pp. 5:1-5:19, 2011 (copyright ACM)

  5. The combination of IDR(s) with BiCGstab(ℓ) is described in: Gerard L.G. Sleijpen and Martin B. van Gijzen, Exploiting BiCGstab(ℓ) strategies to induce dimension reduction. SIAM J. Sci. Comput. Vol. 32, No. 5, pp. 2687-2709, 2010 (copyright SIAM)

  6. A version of IDR(s) that is tuned for parallel and grid computing is described in: T.P. Collignon and M.B. van Gijzen, Minimizing synchronization in IDR(s). Numerical Linear Algebra with Applications, Vol. 18, No. 5, pp. 805–825, 2011 (Copyright John Wiley & Sons, Ltd.)

  7. Flexible and multi-shift IDR variants are described in: M.B. van Gijzen, G.L.G. Sleijpen, and J.P.M Zemke, Flexible and multi‐shift induced dimension reduction algorithms for solving large sparse linear systems. Numerical Linear Algebra with Applications, Vol 22(1), pp. 1–25, 2015

  8. IDR for matrix equations is described in: R. Astudillo and M.B. van Gijzen, Induced Dimension Reduction method for solving linear matrix equations. Procedia Comput. Sci., Vol. 80, pp. 222--232, 2016

  9. An IDR-algorithm for computing eigenpairs is presented in: R. Astudillo and M.B. van Gijzen, A restarted Induced Dimension Reduction method to approximate eigenpairs of large unsymmetric matrices. J. Comput. Appl. Math., Vol. 296, pp. 24--35, 2016.



News and events:

November 2011: IDR(s) has been included in the Collected Algorithms of the ACM as Algorithm 913.

July 8, 2010: Invited talk about IDR(s) at the ICCAM 2010 conference in Leuven, Belgium.

January 2010: IDR(s) (the biortho variant described in [4]) has been included in IFISS 3.0, an open source Incompressible Flow & Iterative Solver Software by Howard Elman, David Silvester and Alison Ramage.

October 27, 2009: Mini symposium "Induced Dimension Reduction (IDR) Methods: a Family of Efficient Krylov Solvers" which was part of the SIAM conference on Applied Linear Algebra LA09.