Lectures on Numerical Mathematics

1. An Outline of the Problems.- x 1.1. Reliability of programs.- x 1.2. The evolution of a program.- x 1.3. Difficulties.- Notes to Chapter 1.- 2. Linear Equations and Inequalities.- x 2.1. The classical algorithm of Gauss.- x 2.2. The triangular decomposition.- x 2.3. Iterative refinement.- x 2.4. Pivoting strategies.- x 2.5. Questions of programming.- x 2.6. The exchange algorithm.- x 2.7. Questions of programming.- x 2.8. Linear inequalities (optimization).- Notes to Chapter 2.- 3. Systems of Equations With Positive Definite Symmetric Coefficient Matrix.- x 3.1. Positive definite matrices.- x 3.2. Criteria for positive definiteness.- x 3.3. The Cholesky decomposition.- x 3.4. Programming the Cholesky decomposition.- x 3.5. Solution of a linear system.- x 3.6. Influence of rounding errors.- x 3.7. Linear systems of equations as a minimum problem.- Notes to Chapter 3.- 4. Nonlinear Equations.- x 4.1. The basic idea of linearization.- x 4.2. Newton's method.- x 4.3. The regula falsi.- x 4.4. Algebraic equations.- x 4.5. Root squaring (Dandelin-Graeffe).- x 4.6. Application of Newton's method to algebraic equations.- Notes to Chapter 4.- 5. Least Squares Problems.- x 5.1. Nonlinear least squares problems.- x 5.2. Linear least squares problems and their classical solution.- x 5.3. Unconstrained least squares approximation through orthogonalization.- x 5.4. Computational implementation of the orthogonalization.- x 5.5. Constrained least squares approximation through orthogonalization.- Notes to Chapter 5.- 6. Interpolation.- x 6.1. The interpolation polynomial.- x 6.2. The barycentric formula.- x 6.3. Divided differences.- x 6.4. Newton's interpolation formula.- x 6.5. Specialization to equidistant xi.- x 6.6. The problematic nature of Newton interpolation.- x 6.7. Hermite interpolation.- x 6.8. Spline interpolation.- x 6.9. Smoothing.- x 6.10.Approximate quadrature.- Notes to Chapter 6.- 7. Approximation.- x 7.1. Critique of polynomial representation.- x 7.2. Definition and basic properties of Chebyshev polynomials.- x 7.3. Expansion in T-polynomials.- x 7.4. Numerical computation of the T-coefficients.- x 7.5. The use of T-expansions.- x 7.6. Best approximation in the sense of Chebyshev (T-approximation).- x 7.7. The Remez algorithm.- Notes to Chapter 7.- 8. Initial Value Problems for Ordinary Differential Equations.- x8.1. Statement of the problem.- x 8.2. The method of Euler.- x 8.3. The order of a method.- x 8.4. Methods of Runge-Kutta type.- x 8.5. Error considerations for the Runge-Kutta method when applied to linear systems of differential equations.- x 8.6. The trapezoidal rule.- x 8.7. General difference formulae.- x 8.8. The stability problem.- x 8.9. Special cases.- Notes to Chapter 8.- 9. Boundary Value Problems For Ordinary Differential Equations.- x 9.1. The shooting method.- x 9.2. Linear boundary value problems.- x 9.3. The Floquet solutions of a periodic differential equation.- x 9.4. Treatment of boundary value problems with difference methods.- x 9.5. The energy method for discretizing continuous problems.- Notes to Chapter 9.- 10. Elliptic Partial Differential Equations, Relaxation Methods.- x10.1. Discretization of the Dirichlet problem.- x10.2. The operator principle.- x10.3. The general principle of relaxation.- x10.4. The method of Gauss-Seidel, overtaxation.- x10.5. The method of conjugate gradients.- x10.6. Application to a more complicated problem.- x10.7. Remarks on norms and the condition of a matrix.- Notes to Chapter 10.- 11. Parabolic and Hyperbolic Partial Differential Equations.- x11.1. One-dimensional heat conduction problems.- x11.2. Stability of the numerical solution.- x11.3. The one-dimensional wave equation.- x11.4. Remarks on two-dimensional heat conduction problems.- Notes to Chapter 11.- 12. The Eigenvalue Problem For Symmetric Matrices.- x12.1. Introduction.- x12.2. Extremal properties of eigenvalues.- x12.3. The classical Jacobi method.- x12.4. Programming considerations.- x12.5. The cyclic Jacobi method.- x12.6. The LR transformation.- x12.7. The LR transformation with shifts.- x12.8. The Householder transformation.- x12.9. Determination of the eigenvalues of a tridiagonal matrix.- Notes to Chapter 12.- 13. The Eigenvalue Problem For Arbitrary Matrices.- x13.1. Susceptibility to errors.- x13.2. Simple vector iteration.- Notes to Chapter 13.- Appendix. An Axiomatic Theory of Numerical Computation with an Application to the Quotient-Difference Algorithm.- Editor's Foreword.- Al. Introduction.- xA1.1. The eigenvalues of a qd-row.- xA1.2. The progressive form of the qd-algorithm.- xA1.3. The generating function of a qd-row.- xA1.4. Positive qd-rows.- xA1.5. Speed of convergence of the qd-algorithm.- xA1.6. The qd-algorithm with shifts.- xA1.7. Deflation after the determination of an eigenvaluec.- A2. Choice of Shifts.- xA2.1. Effect of the shift v on Z'.- xA2.2. Seropositive qd-rows.- x A2.4. A formal algorithm for the determination of eigenvalues.- A3. Finite Arithmetic.- xA3.1. The basic sets.- xA3.2. Properties of the arithmetic.- xA3.3. Monotonicity of the arithmetic.- xA3.4. Precision of the arithmetic.- xA3.5. Underflow and overflow control.- A4. Influence of Rounding Errors.- xA4.1. Persistent properties of the qd-algorithm.- xA4.2. Coincidence.- xA4.3. The differential form of the progressive qd-algorithm.- xA4.4. The influence of rounding errors on convergence.- A5. Stationary Form of the qd-Algorithm.- xA5.1. Development of the algorithm.- xA5.2. The differential form of the stationary qd-algorithm.- xA5.3. Properties of the stationary qd-algorithm.- xA5.4. Safe qd-steps.- Bibliography to the Appendix.- Author Index.