Kincaid D., Cheney W. Numerical Analysis: Mathematics of Scientific Computing

Providence: American Mathematical Society, 2002. — 804 p.This book introduces students with diverse backgrounds to various types of mathematical analysis that are commonly needed in scientific computing. The subject of numerical analysis is treated from a mathematical point of view, offering a complete analysis of methods for scientific computing with appropriate motivations and careful proofs. In an engaging and informal style, the authors demonstrate that many computational procedures and intriguing questions of computer science arise from theorems and proofs. Algorithms are presented in pseudocode, so that students can immediately write computer programs in standard languages or use interactive mathematical software packages. This book occasionally touches upon more advanced topics that are not usually contained in standard textbooks at this level.Numerical Analysis: What Is It?
Basic Concepts and Taylor's Theorem
Orders of Convergence and Additional Basic Concepts
Difference Equations
Floating-Point Numbers and Roundoff Errors
Absolute and Relative Errors: Loss of Significance
Stable and Unstable Computations: ConditioningBisection (Interval Halving) Method
Newton's Method
Secant Method
Fixed Points and Functional Iteration
Computing Roots of Polynomials
Homotopy and Continuation MethodsMatrix Algebra
LU and Cholesky Factorizations
Pivoting and Constructing an Algorithm
Norms and the Analysis of Errors
Neumann Series and Iterative Refinement
Solution of Equations by Iterative Methods
Steepest Descent and Conjugate Gradient Methods
Analysis of Roundoff Error in the Gaussian Algorithm
Review of Basic Concepts
Martix Eigenvalue Problem: Power Method
Schur's and Gershgorin's Theorems
Orthogonal Factorizations and Least-Squares Problems
Singular-Value Decomposition and Pseudo inverses
QR-Algorithm of Francis for the Eigenvalue Problem
Polynomial Interpolation
Divided Differences
Hermite Interpolation
Spline Interpolation
B-Splines: Basic Theory
B-Splines: Applications
Taylor Series
Best Approximation: Least-Squares Theory
Best Approximation: Chebyshev Theory
Interpolation in Higher Dimensions
Continued Fractions
Trigonometric Interpolation
Fast Fourier Transform
Adaptive Approximation
Numerical Differentiation and Richardson Extrapolation
Numerical Integration Based on Interpolation
Gaussian Quadrature
Romberg Integration
Adaptive Quadrature
Sard's Theory of Approximating Functionals
Bernoulli Polynomials and the Euler-Maclaurin Formula
The Existence and Uniqueness of Solutions
Taylor-Series Method
Runge-Kutta Methods
Multistep Methods
Local and Global Errors: Stability
Systems and Higher-Order Ordinary Differential Equations
Boundary-Value Problems
Boundary-Value Problems: Shooting Methods
Boundary-Value Problems: Finite-Differences
Boundary-Value Problems: Collocation
Linear Differential Equations
Stiff Equations
Parabolic Equations: Explicit Methods
Parabolic Equations: Implicit Methods
Problems Without Time Dependence: Finite-Differences
Problems Without Time Dependence: Galerkin Methods
First-Order Partial Differential Equations: Characteristics
Quasilinear Second-Order Equations: Characteristics
Other Methods for Hyperbolic Problems
Multigrid Method
Fast Method s for Poisson's Equation
Convexity and Linear Inequalities
Linear Inequalities
Linear Programming
The Simplex AlgorithmOne-Variable Case
Descent Methods
Analysis of Quadratic Objective Functions
Quadratic-Fitting Algorithms
Nelder-Mead Algorithm
Simulated Annealing
Genetic Algorithms
Convex Programming
Constrained Minimization
Pareto Optimization
Appendix A: An Overview of Mathematical Software