§16§The Eighth EPSRC Numerical Analysis Summer School was held at the Uni versity of Leicester from the 5th to the 17th of July, 1998. This was the third Numerical Analysis Summer School to be held in Leicester. The previous meetings, in 1992 and 1994, had been carefully structured to ensure that each week had a coherent 'theme'. For the 1998 meeting, in order to widen the audience, we decided to relax this constraint. Speakers were chosen to cover what may appear, at first sight, to be quite diverse areas of numeri cal analysis. However, we were pleased with the extent to which the ideas cohered, and particularly enjoyed the discussions which arose from differing interpretations of those ideas. We would like to thank all six of our main speakers for the care which they took in the preparation and delivery of their lectures. In this volume we present their lecture notes in alphabetical rather than chronological order. Nick Higham, Alastair Spence and Nick Trefethen were the speakers i §16§n week 1, while Bernardo Cockburn, Stig Larsson and Bob Skeel were the speakers in week 2. Another new feature of this meeting compared to its predecessors was that we had 'invited seminars'. A numer of established academics based in the UK were asked to participate in the afternoon seminar program. §04§A Simple Introduction to Error Estimation for Nonlinear Hyperbolic Conservation Laws.- 1 Introduction.- 2 Some Convection-Diffusion Problems.- 3 Continuous Dependence for Nonlinear Convection-Diffusion.- 4 Continuous Dependence for Nonlinear Convection.- 5 A Posteriori Error Estimates for Continuous Approximations.- 6 A Posteriori Error Estimates for Discontinuous Approximations.- 7 Concluding Remarks.- Notes on Accuracy and Stability of Algorithms in Numerical Linear Algebra.- 1 Introduction.- 2 Preliminaries.- 3 Symmetric Indefinite Systems.- 4 QR Factorization and Constrained Least Squares Problems.- 5 The Singular Value Decomposition and Jacobi's Method.- Numerical Analysis of Semilinear Parabolic Problems.- 1 The Continuous Problem.- 2 Local a Priori Error Estimates.- 3 Shadowing-First Approach.- 4 A Posteriori Error Estimates.- 5 Shadowing-Second Approach.- Integration Schemes for Molecular Dynamics and Related Applications.- 1 Introduction.- 2 Newtonian Dynamics.- 3 The Lea §04§pfrog Method.- 4 Other Methods.- 5 Multiple Time Steps.- 6 Constrained Dynamics.- 7 Constant-Temperature and Constant-Pressure Ensembles.- 8 Stochastic Dynamics.- Numerical Methods for Bifurcation Problems.- 1 Introduction.- 2 Examples.- 3 Newton's Method and the Implicit Function Theorem.- 4 Computation of Solution Paths.- 5 The Computation of Fold (Turning) Points.- 6 Bifurcation from the Trivial Solution.- 7 Bifurcation in Nonlinear ODEs.- 8 Hopf Bifurcation.- Spectra and Pseudospectra.- 1 Eigenvalues.- 2 Pseudospectra.- 3 A Matrix Example.- 4 An Operator Example.- 5 History of Pseudospectra.
