We revisit real-valued preconditioned iterative methods for the solution of complex linear systems, with an emphasis on symmetric (non-Hermitian) problems. Different choices of the real equivalent formulation are discussed, as well as different types of block preconditioners for Krylov subspace methods. We argue that if either the real or the symmetric part of the coefficient matrix is positive semidefinite, block preconditioners for real equivalent formulations may be a useful alternative to preconditioners for the original complex formulation. Numerical experiments illustrating the performance of the various approaches are presented.

In this paper, a recently introduced block circulant preconditioner for the linear systems of the codes for ordinary differential equations (ODEs) is investigated. Most ODE codes based on implicit formulas, at each integration step, need the solution of one or more unsymmetric linear systems that are often large and sparse. Here, the boundary value methods, a class of implicit methods for the numerical integration of ODEs based on linear multistep formulas, are considered more in detail for initial value problems. Theoretical and practical arguments are given to show that the block circulant preconditioner can give fast preconditioned iterations for various classes of differential problems. Moreover, the P-circulants, a recently introduced circulant approximation for unsymmetric Toeplitz matrices, are shown to be more suitable sometimes than other circulant matrices for the underlying block preconditioner.

Newton–Krylov methods, a combination of Newton-like methods and Krylov subspace methods for solving the Newton equations, often need adequate preconditioning in order to be successful. Approximations of the Jacobian matrices are required to form preconditioners, and this step is very often the dominant cost of Newton–Krylov methods. Therefore, working with preconditioners may destroy the “Jacobian-free” (or matrix-free) setting where the single Jacobian-vector product can be provided without forming and storing the element of the true Jacobian. In this paper, we propose and analyze a preconditioning technique for sequences of nonsymmetric Jacobian matrices based on the update of an earlier preconditioner. The proposed strategy can be implemented in a matrix-free manner. Numerical experiments on popular test problems confirm the effectiveness of the approach in comparison with the standard ILU-preconditioned Newton–Krylov approaches.