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.