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Lectures on Numerical Mathematics

Lectures on Numerical Mathematics

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Lectures on Numerical Mathematics

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Beschreibung

Details

Einband

Taschenbuch

Erscheinungsdatum

17.09.2011

Verlag

Birkhäuser Boston

Seitenzahl

546

Maße (L/B/H)

22.9/15.2/3.1 cm

Beschreibung

Details

Einband

Taschenbuch

Erscheinungsdatum

17.09.2011

Verlag

Birkhäuser Boston

Seitenzahl

546

Maße (L/B/H)

22.9/15.2/3.1 cm

Gewicht

812 g

Auflage

Softcover reprint of the original 1st ed. 1990

Sprache

Englisch

ISBN

978-1-4612-8035-4

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  • Lectures on Numerical Mathematics
  • 1. An Outline of the Problems.-
    1.1. Reliability of programs.-
    1.2. The evolution of a program.-
    1.3. Difficulties.- Notes to Chapter 1.- 2. Linear Equations and Inequalities.-
    2.1. The classical algorithm of Gauss.-
    2.2. The triangular decomposition.-
    2.3. Iterative refinement.-
    2.4. Pivoting strategies.-
    2.5. Questions of programming.-
    2.6. The exchange algorithm.-
    2.7. Questions of programming.-
    2.8. Linear inequalities (optimization).- Notes to Chapter 2.- 3. Systems of Equations With Positive Definite Symmetric Coefficient Matrix.-
    3.1. Positive definite matrices.-
    3.2. Criteria for positive definiteness.-
    3.3. The Cholesky decomposition.-
    3.4. Programming the Cholesky decomposition.-
    3.5. Solution of a linear system.-
    3.6. Influence of rounding errors.-
    3.7. Linear systems of equations as a minimum problem.- Notes to Chapter 3.- 4. Nonlinear Equations.-
    4.1. The basic idea of linearization.-
    4.2. Newton’s method.-
    4.3. The regula falsi.-
    4.4. Algebraic equations.-
    4.5. Root squaring (Dandelin-Graeffe).-
    4.6. Application of Newton’s method to algebraic equations.- Notes to Chapter 4.- 5. Least Squares Problems.-
    5.1. Nonlinear least squares problems.-
    5.2. Linear least squares problems and their classical solution.-
    5.3. Unconstrained least squares approximation through orthogonalization.-
    5.4. Computational implementation of the orthogonalization.-
    5.5. Constrained least squares approximation through orthogonalization.- Notes to Chapter 5.- 6. Interpolation.-
    6.1. The interpolation polynomial.-
    6.2. The barycentric formula.-
    6.3. Divided differences.-
    6.4. Newton’s interpolation formula.-
    6.5. Specialization to equidistant xi.-
    6.6. The problematic nature of Newton interpolation.-
    6.7. Hermite interpolation.-
    6.8. Spline interpolation.-
    6.9. Smoothing.-
    6.10.Approximate quadrature.- Notes to Chapter 6.- 7. Approximation.-
    7.1. Critique of polynomial representation.-
    7.2. Definition and basic properties of Chebyshev polynomials.-
    7.3. Expansion in T-polynomials.-
    7.4. Numerical computation of the T-coefficients.-
    7.5. The use of T-expansions.-
    7.6. Best approximation in the sense of Chebyshev (T-approximation).-
    7.7. The Remez algorithm.- Notes to Chapter 7.- 8. Initial Value Problems for Ordinary Differential Equations.-
    8.1. Statement of the problem.-
    8.2. The method of Euler.-
    8.3. The order of a method.-
    8.4. Methods of Runge-Kutta type.-
    8.5. Error considerations for the Runge-Kutta method when applied to linear systems of differential equations.-
    8.6. The trapezoidal rule.-
    8.7. General difference formulae.-
    8.8. The stability problem.-
    8.9. Special cases.- Notes to Chapter 8.- 9. Boundary Value Problems For Ordinary Differential Equations.-
    9.1. The shooting method.-
    9.2. Linear boundary value problems.-
    9.3. The Floquet solutions of a periodic differential equation.-
    9.4. Treatment of boundary value problems with difference methods.-
    9.5. The energy method for discretizing continuous problems.- Notes to Chapter 9.- 10. Elliptic Partial Differential Equations, Relaxation Methods.-
    10.1. Discretization of the Dirichlet problem.-
    10.2. The operator principle.-
    10.3. The general principle of relaxation.-
    10.4. The method of Gauss-Seidel, overtaxation.-
    10.5. The method of conjugate gradients.-
    10.6. Application to a more complicated problem.-
    10.7. Remarks on norms and the condition of a matrix.- Notes to Chapter 10.- 11. Parabolic and Hyperbolic Partial Differential Equations.-
    11.1. One-dimensional heat conduction problems.-
    11.2. Stability of the numerical solution.-
    11.3. The one-dimensional wave equation.-
    11.4. Remarks on two-dimensional heat conduction problems.- Notes to Chapter 11.- 12. The Eigenvalue Problem For Symmetric Matrices.-
    12.1. Introduction.-
    12.2. Extremal properties of eigenvalues.-
    12.3. The classical Jacobi method.-
    12.4. Programming considerations.-
    12.5. The cyclic Jacobi method.-
    12.6. The LR transformation.-
    12.7. The LR transformation with shifts.-
    12.8. The Householder transformation.-
    12.9. Determination of the eigenvalues of a tridiagonal matrix.- Notes to Chapter 12.- 13. The Eigenvalue Problem For Arbitrary Matrices.-
    13.1. Susceptibility to errors.-
    13.2. 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.-
    A1.1. The eigenvalues of a qd-row.-
    A1.2. The progressive form of the qd-algorithm.-
    A1.3. The generating function of a qd-row.-
    A1.4. Positive qd-rows.-
    A1.5. Speed of convergence of the qd-algorithm.-
    A1.6. The qd-algorithm with shifts.-
    A1.7. Deflation after the determination of an eigenvaluec.- A2. Choice of Shifts.-
    A2.1. Effect of the shift v on Z’.-
    A2.2. Seropositive qd-rows.-
    A2.4. A formal algorithm for the determination of eigenvalues.- A3. Finite Arithmetic.-
    A3.1. The basic sets.-
    A3.2. Properties of the arithmetic.-
    A3.3. Monotonicity of the arithmetic.-
    A3.4. Precision of the arithmetic.-
    A3.5. Underflow and overflow control.- A4. Influence of Rounding Errors.-
    A4.1. Persistent properties of the qd-algorithm.-
    A4.2. Coincidence.-
    A4.3. The differential form of the progressive qd-algorithm.-
    A4.4. The influence of rounding errors on convergence.- A5. Stationary Form of the qd-Algorithm.-
    A5.1. Development of the algorithm.-
    A5.2. The differential form of the stationary qd-algorithm.-
    A5.3. Properties of the stationary qd-algorithm.-
    A5.4. Safe qd-steps.- Bibliography to the Appendix.- Author Index.