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A Theoretical Introduction to Numerical Analysis





ISBN 9780367453398
Published December 17, 2019 by Chapman and Hall/CRC
552 Pages 50 B/W Illustrations

 
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Book Description

A Theoretical Introduction to Numerical Analysis presents the general methodology and principles of numerical analysis, illustrating these concepts using numerical methods from real analysis, linear algebra, and differential equations. The book focuses on how to efficiently represent mathematical models for computer-based study.

An accessible yet rigorous mathematical introduction, this book provides a pedagogical account of the fundamentals of numerical analysis. The authors thoroughly explain basic concepts, such as discretization, error, efficiency, complexity, numerical stability, consistency, and convergence. The text also addresses more complex topics like intrinsic error limits and the effect of smoothness on the accuracy of approximation in the context of Chebyshev interpolation, Gaussian quadratures, and spectral methods for differential equations. Another advanced subject discussed, the method of difference potentials, employs discrete analogues of Calderon’s potentials and boundary projection operators. The authors often delineate various techniques through exercises that require further theoretical study or computer implementation.

By lucidly presenting the central mathematical concepts of numerical methods, A Theoretical Introduction to Numerical Analysis provides a foundational link to more specialized computational work in fluid dynamics, acoustics, and electromagnetism.

Table of Contents

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

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Author(s)

Biography

Victor S. Ryaben'kii, Semyon V Tsynkov

Reviews

“… presents the general methodology and principles of numerical analysis, illustrating the key concepts using numerical methods from real analysis, linear algebra, and differential equations. The book focuses on hoe to efficiently represent mathematical models for computer-based study. … this book provides a pedagogical account of the fundamentals of numerical analysis. … provides a foundation link to more specialized computational work in mathematics, science, and engineering. … Discusses three common numerical areas: interpolation and quadratures, linear and nonlinear solvers, and finite differences. Explains the most fundamental and universal concepts, including error, efficiency, complexity, stability, and convergence. Addresses advance topics, such as intrinsic accuracy limits, saturation of numerical methods by smoothness, and the method of difference potentials. Provides rigorous proofs for all important mathematical results. Includes numerous examples and exercises to illustrate key theoretical ideas and to enable independent study. ”
— In Mathematical Reviews, Issue 2007g


“It is an excellent book, having a wide spectrum of classical and advanced topics. The book has all the advantages of the Russian viewpoint as well as the Western one.”
—David Gottlieb, Brown University, Providence, Rhode Island, USA