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.
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Version of August 2010. This is the
bi-ortho variant of IDR(s) (with enhancements) that is described in
 (see below).
The most important changes with respect to version of December 2008 are:
Preconditioner can be passed in decomposed form;
Matrix-vector multiplication and preconditioning operations can be defined by functions;
Residual smoothing (optional);
Residual replacements to achieve accuracy close to machine precision (optional).
example_idrs.m (needs idrs.m).
This MATLAB script defines a 3D discretised convection-diffusion-reaction problem on the unit cube. The parameters can be changed via a user interface to create a different test problem. The problem is solved with IDR(1), IDR(2), IDR(4), IDR(8), and with the built-in MATLAB routines for (full) GMRES and Bi-CGSTAB. The picture below shows the convergence of the methods for the default parameters, which specify a highly non-symmetric and indefinite problem consisting of about 60,000 equations.
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).
idrs_f90.tar: This file contains an advanced F90/F95 implementation of IDR(s) for linear systems and for linear matrix equations. It comes with a test program that explains how to use the code. Feedback is welcome!
idrs_Arash.tar: this file contains a simple F90/F95 implementation of IDR(s). It has been written and made available by Arash Ghasemi (National Center for Computational Engineering, University of Tennessee).
Moritz Schauer has added IDR(s) to the Julia package https://github.com/JuliaMath/IterativeSolvers.jl.
Calling IDR(s) is as simple as:
julia> using IterativeSolvers
julia> A = sprand(10_000, 10_000, 10 / 10_000) + 3I
b = A * x
xhat = IterativeSolvers.idrs(A, b)
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)
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
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)
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)
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)
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.)
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
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
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.
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 ) 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.