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