1st Edition

# A Theoretical Introduction to Numerical Analysis

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**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.

**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.

**Factorization**

*QR*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

### Biography

Victor S. Ryaben'kii, Semyon V Tsynkov

“… 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