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

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 $s$ the codimension of a fixed subspace. We describe the algorithm and the underlying theory and present numerical experiments to illustrate the theoretical properties of the method and its performance for systems arising from different applications. Our experiments show that IDR($s$) is competitive with or superior to most Bi-CG-based methods and outperforms Bi-CGSTAB when $s > 1$.