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| Preface | |
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| Solution of equations by iteration | |
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| Introduction | |
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| Simple iteration | |
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| Iterative solution of equations | |
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| Relaxation and Newton's method | |
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| The secant method | |
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| The bisection method | |
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| Global behaviour | |
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| Notes | |
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| Exercises | |
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| Solution of systems of linear equations | |
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| Introduction | |
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| Gaussian elimination | |
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| LU factorisation | |
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| Pivoting | |
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| Solution of systems of equations | |
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| Computational work | |
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| Norms and condition numbers | |
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| Hilbert matrix | |
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| Least squares method | |
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| Notes | |
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| Exercises | |
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| Special matrices | |
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| Introduction | |
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| Symmetric positive definite matrices | |
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| Tridiagonal and band matrices | |
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| Monotone matrices | |
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| Notes | |
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| Exercises | |
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| Simultaneous nonlinear equations | |
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| Introduction | |
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| Simultaneous iteration | |
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| Relaxation and Newton's method | |
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| Global convergence | |
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| Notes | |
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| Exercises | |
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| Eigenvalues and eigenvectors of a symmetric matrix | |
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| Introduction | |
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| The characteristic polynomial | |
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| Jacobi's method | |
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| The Gerschgorin theorems | |
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| Householder's method | |
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| Eigenvalues of a tridiagonal matrix | |
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| The QR algorithm | |
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| The QR factorisation revisited | |
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| The definition of the QR algorithm | |
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| Inverse iteration for the eigenvectors | |
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| The Rayleigh quotient | |
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| Perturbation analysis | |
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| Notes | |
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| Exercises | |
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| Polynomial interpolation | |
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| Introduction | |
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| Lagrange interpolation | |
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| Convergence | |
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| Hermite interpolation | |
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| Differentiation | |
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| Notes | |
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| Exercises | |
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| Numerical integration-I | |
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| Introduction | |
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| Newton-Cotes formulae | |
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| Error estimates | |
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| The Runge phenomenon revisited | |
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| Composite formulae | |
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| The Euler-Maclaurin expansion | |
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| Extrapolation methods | |
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| Notes | |
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| Exercises | |
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| Polynomial approximation in the [infinity]-norm | |
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| Introduction | |
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| Normed linear spaces | |
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| Best approximation in the [infinity]-norm | |
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| Chebyshev polynomials | |
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| Interpolation | |
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| Notes | |
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| Exercises | |
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| Approximation in the 2-norm | |
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| Introduction | |
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| Inner product spaces | |
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| Best approximation in the 2-norm | |
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| Orthogonal polynomials | |
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| Comparisons | |
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| Notes | |
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| Exercises | |
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| Numerical integration - II | |
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| Introduction | |
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| Construction of Gauss quadrature rules | |
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| Direct construction | |
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| Error estimation for Gauss quadrature | |
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| Composite Gauss formulae | |
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| Radau and Lobatto quadrature | |
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| Note | |
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| Exercises | |
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| Piecewise polynomial approximation | |
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| Introduction | |
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| Linear interpolating splines | |
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| Basis functions for the linear spline | |
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| Cubic splines | |
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| Hermite cubic splines | |
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| Basis functions for cubic splines | |
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| Notes | |
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| Exercises | |
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| Initial value problems for ODEs | |
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| Introduction | |
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| One-step methods | |
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| Consistency and convergence | |
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| An implicit one-step method | |
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| Runge-Kutta methods | |
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| Linear multistep methods | |
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| Zero-stability | |
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| Consistency | |
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| Dahlquist's theorems | |
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| Systems of equations | |
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| Stiff systems | |
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| Implicit Runge-Kutta methods | |
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| Notes | |
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| Exercises | |
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| Boundary value problems for ODEs | |
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| Introduction | |
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| A model problem | |
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| Error analysis | |
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| Boundary conditions involving a derivative | |
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| The general self-adjoint problem | |
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| The Sturm-Liouville eigenvalue problem | |
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| The shooting method | |
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| Notes | |
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| Exercises | |
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| The finite element method | |
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| Introduction: the model problem | |
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| Rayleigh-Ritz and Galerkin principles | |
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| Formulation of the finite element method | |
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| Error analysis of the finite element method | |
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| A posteriori error analysis by duality | |
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| Notes | |
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| Exercises | |
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| An overview of results from real analysis | |
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| WWW-resources | |
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| Bibliography | |
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| Index | |