Numerical Analysis - 1st Edition - ISBN: 9780125285605, 9781483268507

Numerical Analysis

1st Edition

A Second Course

Authors: James M. Ortega
Editors: Werner Rheinboldt
eBook ISBN: 9781483268507
Imprint: Academic Press
Published Date: 1st January 1972
Page Count: 214
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Description

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

Details

No. of pages:
214
Language:
English
Copyright:
© Academic Press 1972
Published:
Imprint:
Academic Press
eBook ISBN:
9781483268507

About the Author

James M. Ortega

About the Editor

Werner Rheinboldt

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