... In this paper, we present a novel BCG-like approach, the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported.

this paper is as follows. In Section 2, we present some background material on general Krylov subspace methods, of which CGtype algorithms are a special case. We recall the outstanding properties of CG and discuss the issue of optimal extensions of CG to non-Hermitian matrices. We also review GMRES and related methods, as well as CG-like algorithms for the special case of Hermitian indefinite linear systems. Finally, we briefly discuss the basic idea of preconditioning. In Section 3, we turn to Lanczos-based iterative methods for general non-Hermitian linear systems. First, we consider the nonsymmetric Lanczos process, with particular emphasis on the possible breakdowns and potential instabilities in the classical algorithm. Then we describe recent advances in understanding these problems and overcoming them by using look-ahead techniques. Moreover, we describe the quasi-minimal residual algorithm (QMR) proposed by Freund and Nachtigal (1990), which uses the look-ahead Lanczos process to obtain quasi-optimal approximate solutions. Next, a survey of transposefree Lanczos-based methods is given. We conclude this section with comments on other related work and some historical remarks. In Section 4, we elaborate on CGNR and CGNE and we point out situations where these approaches are optimal. The general class of Krylov subspace methods also contains parameter-dependent algorithms that, unlike CG-type schemes, require explicit information on the spectrum of the coefficient matrix. In Section 5, we discuss recent insights in obtaining appropriate spectral information for parameter-dependent Krylov subspace methods. After that, 4 R.W. Freund, G.H. Golub and N.M. Nachtigal

We consider conjugate gradient type methods for the solution of large linear systems Az = b with complex coefficient matrices of the type A = T + io1 where T is Hermitian and u a real scalar. Three different conjugate gradient type approaches with it-erates defined by a minimal residual property, a Galerkin type condition, and an Euclidian error minimization, respectively, are investigated. In particular, we propose numerically stable implementations based on the ideas behind Paige and Saunders’s SYMMLQ and MINRES for real symmetric matrices and derive error bounds for all three methods. It is shown how the special shift structure of A can be preserved by using polynomial precon-ditioning, and results on the optimal choice of the polynomial preconditioner are given. Also, we report on some numerical experiments for matrices arising from finite difference approximations to the complex Helmholtz equation.

Compared to Krylov space methods based on orthogonal or oblique projection, the Chebyshev iteration does not require inner products and is therefore particularly suited for massively parallel computers with high communication cost. Here, six different algorithms that implement this method are presented and compared with respect to roundoff effects, in particular, the ultimately achievable accuracy. Two of these algorithms replace the three-term recurrences by more accurate coupled two-term recurrences and seem to be new. It is also shown that, for real data, the classical three-term Chebyshev iteration is never seriously affected by roundoff, in contrast to the corresponding version of the conjugate gradient method. Even for complex data, strongroundoff effects are seen to be limited to very special situations where convergence is anyway slow. The Chebyshev iteration is applicable to symmetric definite linear systems and to nonsymmetric matrices whose eigenvalues are known to be confined to an elliptic domain that does not include the origin. Also considered is a corresponding stationary 2-step method, which has the same asymptotic convergence behavior and is additionally suitable for mildly nonlinear problems.

When solving PDE's by means of numerical methods one often has to deal with large systems of linear equations, specifically if the PDE is time-independent or if the time-integrator is implicit. For real life problems, these large systems can often only be solved by means of some iterative method. Even if the systems are preconditioned, the basic iterative method often converges slowly or even diverges. We discuss and classify algebraic techniques to accelerate the basic iterative method. Our discussion includes methods like CG, GCR, ORTHODIR, GMRES, CGNR, Bi-CG and their modifications like GMRESR, CG-S, BiCGSTAB. We place them in a frame, discuss their convergence behavior and their advantages and drawbacks. 1 Introduction Our aim is to compute acceptable approximations for the solution x of the equation Ax = b; (1) where A and b are given, A is a non-singular n \Theta n-matrix, A is sparse, n is large and b an n-vector. We will assume A and b to be real, but our methods are easily ...