Numerical Analysis
A Second Course
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Computer Science and Applied Mathematics: Numerical Analysis: A Second Course presents some of the basic theoretical results pertaining to the three major problem areas of numerical analysis—rounding error, discretization error, and convergence error. This book is organized into four main topics: mathematical stability and ill conditioning, discretization error, convergence of iterative methods, and rounding error. In these topics, this text specifically discusses the systems of linear algebraic equations, eigenvalues and eigenvectors, and differential and difference equations. The discretization error for initial and boundary value problems, systems of linear and nonlinear equations, and rounding error for Gaussian elimination are also elaborated. This publication is recommended for undergraduate level students and students taking a one-semester first-year graduate course for computer science and mathematics majors.
Table of Contents
Preface
List of Commonly Used Symbols
Introduction
Chapter 1 Linear Algebra
1.1 Eigenvalues and Canonical Forms
1.2 Vector Norms
1.3 Matrix Norms
Part I Mathematically and Ill Conditioning
Chapter 2 Systems of Linear Algebraic Equations
2.1 Basic Error Estimates and Condition Numbers
2.2 A Posteriori Bounds and Eigenvector Computations
Chapter 3 Eigenvalues and Eigenvectors
3.1 Continuity Results
3.2 The Gerschgorin and Bauer-Fike Theorems
3.3 Special Results for Symmetric Matrices
Chapter 4 Differential and Difference Equations
4.1 Differential Equations
4.2 Difference Equations
Part II Discretization Error
Chapter 5 Discretization Error for Initial Value Problems
5.1 Consistency and Stability
5.2 Convergence and Order
Chapter 6 Discretization Error for Boundary Value Problems
6.1 The Maximum Principle
6.2 Matrix Methods
Part III Convergence of Iterative Methods
Chapter 7 Systems of Linear Equations
7.1 Convergence
7.2 Rate of Convergence
7.3 Applications to Differential Equations
Chapter 8 Systems of Nonlinear Equations
8.1 Local Convergence and Rate of Convergence
8.2 Error Estimates
8.3 Global Convergence
Part IV Rounding Error
Chapter 9 Rounding Error for Gaussian Elimination
9.1 Review of the Method
9.2 Rounding Error and Interchange Strategies
9.3 Backward Error Analysis
9.4 Iterative Refinement
Bibliography
Index
Product details
- No. of pages: 214
- Language: English
- Copyright: © Academic Press 1972
- Published: January 1, 1972
- Imprint: Academic Press
- eBook ISBN: 9781483268507