IDR: A new generation of Krylov subspace methods?

The induced dimension reduction (IDR) technique developed by Sonneveld and van Gijzen [1] is a powerful concept resulting in a variety of transpose-free Krylov subspace methods based on short-term recurrences. We present the main differences between and similarities of IDR methods and classical Krylov subspace methods; our tool of trade is the so-called generalized Hessenberg decomposition. The concept of “transfer” of techniques from the setting of (classical) Krylov subspace methods to the IDR based methods is introduced. For simplicity, we only sketch some recent results in the fields of eigenvalue computations and of solution of linear systems.