Lectures on Numerical Mathematics

Lectures on Numerical Mathematics by H. Rutishauser.

The present book is an edition of the manuscripts to the courses "Numerical Methods I" and "Numerical Mathematics I and II" which Professor H. Rutishauser held at the E.T.H. in Zurich. The first-named course was newly conceived in the spring semester of 1970, and intended for beg...

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Bibliographic Details
Main Author: Rutishauser, H. (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser, 1990.
Edition:1st ed. 1990.
Series:Springer eBook Collection.
Subjects:
Numerical analysis.
Computer mathematics.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 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.

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