A Theoretical Introduction to Numerical Analysis Book

PREFACE
ACKNOWLEDGMENTS
INTRODUCTION
Discretization Conditioning Error On Methods of Computation
INTERPOLATION OF FUNCTIONS. QUADRATURES
ALGEBRAIC INTERPOLATION
Existence and Uniqueness of Interpolating Polynomial Classical Piecewise Polynomial Interpolation Smooth Piecewise Polynomial Interpolation (Splines)
Interpolation of Functions of Two Variables
TRIGONOMETRIC INTERPOLATION
Interpolation of Periodic Functions Interpolation of Functions on an Interval. Relation between Algebraic and Trigonometric Interpolation
COMPUTATION OF DEFINITE INTEGRALS. QUADRATURES
Trapezoidal Rule, Simpson’s Formula, and the Like Quadrature Formulae with No Saturation. Gaussian Quadratures Improper Integrals. Combination of Numerical and Analytical Methods Multiple Integrals
SYSTEMS OF SCALAR EQUATIONS
SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS: DIRECT METHODS
Different Forms of Consistent Linear Systems Linear Spaces, Norms, and Operators Conditioning of Linear Systems Gaussian Elimination and Its Tri-Diagonal Version Minimization of Quadratic Functions and Its Relation to Linear Systems The Method of Conjugate Gradients Finite Fourier Series
ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS
Richardson Iterations and the Like Chebyshev Iterations and Conjugate Gradients Krylov Subspace Iterations Multigrid Iterations
OVERDETERMINED LINEAR SYSTEMS. THE METHOD OF LEAST SQUARES
Examples of Problems that Result in Overdetermined Systems Weak Solutions of Full Rank Systems. QR Factorization Rank Deficient Systems. Singular Value Decomposition
NUMERICAL SOLUTION OF NONLINEAR EQUATIONS AND SYSTEMS
Commonly Used Methods of Rootfinding Fixed Point Iterations Newton’s Method
THE METHOD OF FINITE DIFFERENCES FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS
NUMERCAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS
Examples of Finite-Difference Schemes. Convergence Approximation of Continuous Problem by a Difference Scheme. Consistency Stability of Finite-Difference Schemes The Runge-Kutta Methods Solution of Boundary Value Problems Saturation of Finite-Difference Methods The Notion of Spectral Methods
FINITE-DIFFERENCE SCHEMES FOR PARTIAL DIFFERENTIAL EQUATIONS
Key Definitions and Illustrating Examples Construction of Consistent Difference Schemes Spectral Stability Criterion for Finite-Difference Cauchy Problems Stability for Problems with Variable Coefficients Stability for Initial Boundary Value Problems Explicit and Implicit Schemes for the Heat Equation
DISCONTINUOUS SOLUTIONS AND METHODS OF THEIR COMPUTATION
Differential Form of an Integral Conservation Law Construction of Difference Schemes
DISCRETE METHODS FOR ELLIPTIC PROBLEMS
A Simple Finite-Difference Scheme. The Maximum Principle The Notion of Finite Elements. Ritz and Galerkin Approximations
THE METHODS OF BOUNDARY EQUATIONS FOR THE NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS
BOUNDARY INTEGRAL EQUATIONS AND THE METHOD OF BOUNDARY ELEMENTS
Reduction of Boundary Value Problems to Integral Equations Discretization of Integral Equations and Boundary Elements The Range of Applicability for Boundary Elements
BOUNDARY EQUATIONS WITH PROJECTIONS AND THE METHOD OF DIFFERENCE POTENTIALS
Formulation of Model Problems Difference Potentials Solution of Model Problems
LIST OF FIGURES
REFERENCED BOOKS
REFERENCED JOURNAL ARTICLES
INDEX