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Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This new edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems. Table of Contents Preface to the first edition Preface to the second edition Flowchart of contents Part I. Ordinary differential equations: 1. Euler's method and beyond 2. Multistep methods 3. Runge–Kutta methods 4. Stiff equations 5. Geometric numerical integration 6. Error control 7. Nonlinear algebraic systems Part II. The Poisson equation: 8. Finite difference schemes 9. The finite element method 10. Spectral methods 11. Gaussian elimination for sparse linear equations 12. Classical iterative methods for sparse linear equations 13. Multigrid techniques 14. Conjugate gradients 15. Fast Poisson solvers Part III. Partial differential equations of evolution: 16. The diffusion equation 17. Hyperbolic equations Appendix. Bluffer's guide to useful mathematics: A.1. Linear algebra A.2. Analysis Bibliography Index