# 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.

## MATLAB code:

• idrs.m Version of August 2010. This is the bi-ortho variant of IDR(s) (with enhancements) that is described in [4] (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).

• test_idrs.tgz. A testset of 11 examples, also includes idrs.m. This manual describes idrs.m and the accompanying testset.

• 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.

## 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 linear systems. SIAM J. Sci. Comput. Vol. 31, No. 2, pp. 1035-1062, 2008 (copyright SIAM)

2. 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)

3. 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)

4. 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)

5. 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.)

## 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.