The 3-term Lanczos process leads, for a symmetric matrix, to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of symmetric indefinite linear systems, by solving the reduced system in one way or another. This leads to well-known methods: MINRES, GMRES, and SYMMLQ. We will discuss in what way and to what extent these approaches differ in their sensitivity to rounding errors. In our analysis we will assume that the Lanczos basis is generated in exactly the same way for the different methods, and we will not consider the errors in the Lanczos process itself. We will show that the method of solution may lead, under certain circumstances, to large additional errors, that are not corrected by continuing the iteration process. Our findings are supported and illustrated by numerical examples. 1 Introduction We will consider iterative methods for the construction of approximate solutions, starting with...

apier nach 1 ISO 9706 Contents 1 Introduction 7 1.1 What is a PIM? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Different types of PIMs . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Organization and summary of our results . . . . . . . . . . . . . 9 2 Background 13 2.1 Krylov spaces and the Arnoldi process . . . . . . . . . . . . . . . 13 2.2 Exterior mapping functions and Faber polynomials . . . . . . . . 14 2.3 Inclusion sets and asymptotic analysis . . . . . . . . . . . . . . . 15 3 Inclusion sets generated by the conformal 'bratwurst' maps 19 3.1 Derivation of the maps . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Definition and properties of the 'bratwurst' shape sets . . . . . . 23 3.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . 25 4 The hybrid ABF method for non-hermitian linear systems 29 4.1 Faber polynomials for the inclusion sets