Mathematical Review and Computer Arithmetic
Mathematical Review
Computer Arithmetic
Interval Computations
Numerical Solution of Nonlinear Equations of One Variable
Introduction
Bisection Method
The Fixed Point Method
Newton’s Method (Newton–Raphson Method)
The Univariate Interval Newton Method
Secant Method and Müller’s Method
Aitken Acceleration and Steffensen’s Method
Roots of Polynomials
Additional Notes and Summary
Numerical Linear Algebra
Basic Results from Linear Algebra
Normed Linear Spaces
Direct Methods for Solving Linear Systems
Iterative Methods for Solving Linear Systems
The Singular Value Decomposition
Approximation Theory
Introduction
Norms, Projections, Inner Product Spaces, and Orthogonalization in Function Spaces
Polynomial Approximation
Piecewise Polynomial Approximation
Trigonometric Approximation
Rational Approximation
Wavelet Bases
Least Squares Approximation on a Finite Point Set
Eigenvalue-Eigenvector Computation
Basic Results from Linear Algebra
The Power Method
The Inverse Power Method
Deflation
The QR Method
Jacobi Diagonalization (Jacobi Method)
Simultaneous Iteration (Subspace Iteration)
Numerical Differentiation and Integration
Numerical Differentiation
Automatic (Computational) Differentiation
Numerical Integration
Initial Value Problems for Ordinary Differential Equations
Introduction
Euler’s Method
Single-Step Methods: Taylor Series and Runge–Kutta
Error Control and the Runge–Kutta–Fehlberg Method
Multistep Methods
Predictor-Corrector Methods
Stiff Systems
Extrapolation Methods
Application to Parameter Estimation in Differential Equations
Numerical Solution of Systems of Nonlinear Equations
Introduction and Fréchet Derivatives
Successive Approximation (Fixed Point Iteration) and the Contraction Mapping Theorem
Newton’s Method and Variations
Multivariate Interval Newton Methods
Quasi-Newton Methods (Broyden’s Method)
Methods for Finding All Solutions
Optimization
Local Optimization
Constrained Local Optimization
Constrained Optimization and Nonlinear Systems
Linear Programming
Dynamic Programming
Global (Non-Convex) Optimization
Boundary Value Problems and Integral Equations
Boundary Value Problems
Approximation of Integral Equations
Appendix: Solutions to Selected Exercises
References
Index
Exercises appear at the end of each chapter.