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A Theoretical Introduction to Numerical Analysis

Victor S. Ryaben'kii and Semyon V. Tsynkov
Publisher: 
Chapman & Hall/CRC
Publication Date: 
2007
Number of Pages: 
537
Format: 
Hardcover
Price: 
79.95
ISBN: 
1584886072
Category: 
Textbook
We do not plan to review this 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

Dummy View - NOT TO BE DELETED