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This book is primarily intended for undergraduates in mathematics, the physical sciences and engineering. It introduces students to most of the techniques forming the core component of courses in numerical analysis. The text is divided into eight chapters which are largely self-contained. However, with a subject as intricately woven as mathematics, there is inevitably some interdependence between them. The level of difficulty varies and, although emphasis is firmly placed on the methods themselves rather than their analysis, we have not hesitated to include theoretical material when we consider it to be sufficiently interesting. However, it should be possible to omit those parts that do seem daunting while still being able to follow the worked examples and to tackle the exercises accompanying each section. Familiarity with the basic results of analysis and linear algebra is assumed since these are normally taught in first courses on mathematical methods. For reference purposes a list of theorems used in the text is given in the appendix.

Numerical Analysis

1 Introduction.- 1.1 Rounding errors and instability.- 2 Linear algebraic equations.- 2.1 Gauss elimination.- 2.2 Matrix decomposition methods.- 2.3 Iterative methods.- 3 Non-linear algebraic equations.- 3:1 Bracketing methods.- 3.2 Fixed point iteration.- 3.3 Newton's method.- 3.4 Systems of non-linear equations.- 4 Eigenvalues and eigenvectors.- 4.1 The power method.- 4.2 Deflation.- 4.3 Jacobi's method.- 4.4 Sturm sequence iteration.- 4.5 Givens' and Householder's methods.- 4.6 The LR and QR methods.- 4.7 Hessenberg form.- 5 Methods of approximation theory.- 5.1 Polynomial interpolation: Lagrange form.- 5.2 Polynomial interpolation: divided difference form.- 5.3 Polynomial interpolation: finite difference form.- 5.4 Hermite interpolation.- 5.5 Cubic spline interpolation.- 5.6 Least squares approximation to discrete data.- 5.7 Least squares approximation to continuous functions.- 6 Numerical differentiation and integration.- 6.1 Numerical differentiation.- 6.2 Numerical integration: Newton-Cotes formulas.- 6.3 Quadrature rules in composite form.- 6.4 Romberg's method.- 6.5 Simpson's adaptive quadrature.- 6.6 Gaussian quadrature.- 7 Ordinary differential equations: initial value problems.- 7.1 Derivation of linear multistep methods.- 7.2 Analysis of linear multistep methods.- 7.3 Runge-Kutta methods.- 7.4 Systems and higher order equations.- 8 Ordinary differential equations: boundary value problems.- 8.1 The finite difference method.- 8.2 The shooting method.- References.- Solutions to exercises.