A Theoretical Introduction to Numerical Analysis: 1st Edition (Hardback) book cover

A Theoretical Introduction to Numerical Analysis

1st Edition

By Victor S. Ryaben'kii, Semyon V. Tsynkov

Chapman and Hall/CRC

552 pages | 50 B/W Illus.

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pub: 2006-11-02
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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.

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

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

Subject Categories

BISAC Subject Codes/Headings:
MAT021000
MATHEMATICS / Number Systems
SCI040000
SCIENCE / Mathematical Physics
TEC009070
TECHNOLOGY & ENGINEERING / Mechanical