An Introduction to Numerical Methods and Analysis / Edition 2

An Introduction to Numerical Methods and Analysis / Edition 2

by James F. Epperson
     
 

Praise for the First Edition
". . . outstandingly appealing with regard to its style, contents, considerations of requirements of practice, choice of examples, and exercises."
Zentralblatt MATH

". . . carefully structured with many detailed worked examples."
The Mathematical Gazette

The Second Edition

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Overview

Praise for the First Edition
". . . outstandingly appealing with regard to its style, contents, considerations of requirements of practice, choice of examples, and exercises."
Zentralblatt MATH

". . . carefully structured with many detailed worked examples."
The Mathematical Gazette

The Second Edition of the highly regarded An Introduction to Numerical Methods and Analysis provides a fully revised guide to numerical approximation. The book continues to be accessible and expertly guides readers through the many available techniques of numerical methods and analysis.

An Introduction to Numerical Methods and Analysis, Second Edition reflects the latest trends in the field, includes new material and revised exercises, and offers a unique emphasis on applications. The author clearly explains how to both construct and evaluate approximations for accuracy and performance, which are key skills in a variety of fields. A wide range of higher-level methods and solutions, including new topics such as the roots of polynomials, spectral collocation, finite element ideas, and Clenshaw-Curtis quadrature, are presented from an introductory perspective, and the Second Edition also features:

  • Chapters and sections that begin with basic, elementary material followed by gradual coverage of more advanced material
  • Exercises ranging from simple hand computations to challenging derivations and minor proofs to programming exercises
  • Widespread exposure and utilization of MATLAB®
  • An appendix that contains proofs of various theorems and other material

The book is an ideal textbook for students in advanced undergraduate mathematics and engineering courses who are interested in gaining an understanding of numerical methods and numerical analysis.

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Product Details

ISBN-13:
9781118367599
Publisher:
Wiley
Publication date:
10/07/2013
Edition description:
New Edition
Pages:
614
Product dimensions:
6.90(w) x 10.00(h) x 1.30(d)

Table of Contents

Preface
Acknowledgments
Ch. 1Introductory Concepts and Calculus Review1
1.1Basic Tools of Calculus2
1.2Error, Approximate Equality, and Asymptotic Order Notation14
1.3A Primer on Computer Arithmetic20
1.4A Word on Computer Languages and Software28
1.5Simple Approximations29
1.6Application: Approximating the Natural Logarithm33
Ch. 2A Survey of Simple Methods and Tools37
2.1Homer's Rule and Nested Multiplication37
2.2Difference Approximations to the Derivative40
2.3Application: Euler's Method for Initial Value Problems48
2.4Linear Interpolation53
2.5Application: The Trapezoid Rule60
2.6Solution of Tridiagonal Linear Systems69
2.7Application: Simple Two-Point Boundary Value Problems75
Ch. 3Root Finding80
3.1The Bisection Method81
3.2Newton's Method: Derivation and Examples87
3.3How to Stop Newton's Method93
3.4Application: Division Using Newton's Method96
3.5The Newton Error Formula100
3.6Newton's Method: Theory and Convergence105
3.7Application: Computation of the Square Root109
3.8The Secant Method: Derivation and Example111
3.9Fixed-Point Iteration116
3.10Special Topics in Root-Finding Methods126
Ch. 4Interpolation and Approximation150
4.1Lagrange Interpolation151
4.2Newton Interpolation and Divided Differences156
4.3Interpolation Error166
4.4Application: Muller's Method and Inverse Quadratic170
4.5Application: More Approximation to the Derivative174
4.6Hermite Interpolation177
4.7Piecewise Polynomial Interpolation182
4.8An Introduction to Splines189
4.9Application: Solution of Boundary Value Problems204
4.10Least Squares Concepts in Approximation209
4.11Advanced Topics in Interpolation Error230
4.12Literature and Software Discussion243
Ch. 5Numerical Integration245
5.1A Review of the Definite Integral246
5.2Improving the Trapezoid Rule248
5.3Simpson's Rule and Degree of Precision253
5.4The Midpoint Rule265
5.5Application: Stirling's Formula268
5.6Gaussian Quadrature270
5.7Extrapolation Methods281
5.8Special Topics in Numerical Integration288
Ch. 6Numerical Methods for Ordinary Differential Equations312
6.1The Initial Value Problem: Background313
6.2Euler's Method318
6.3Analysis of Euler's Method322
6.4Variants of Euler's Method326
6.5Single Step Methods: Runge-Kutta343
6.6Multistep Methods350
6.7Stability Issues356
6.8Application to Systems of Equations363
6.9Adaptive Solvers370
6.10Boundary Value Problems383
6.11Literature and Software Discussion392
Ch. 7Numerical Methods for the Solution of Systems394
7.1Linear Algebra Review395
7.2Linear Systems and Gaussian Elimination397
7.3Operation Counts404
7.4The LU Factorization406
7.5Perturbation, Conditioning, and Stability416
7.6SPD Matrices and the Cholesky Decomposition434
7.7Iterative Methods for Linear Systems: A Brief Survey437
7.8Nonlinear Systems: Newton's Method and Related Ideas446
7.9Application: Numerical Solution of Nonlinear BVPs452
7.10Literature and Software Discussion454
Ch. 8Approximate Solution of the Algebraic Eigenvalue Problem456
8.1Eigenvalue Review457
8.2Reduction to Hessenberg Form463
8.3Power Methods471
8.4An Overview of the QR Iteration490
Ch. 9A Survey of Finite Difference Methods for Partial Differential Equations500
9.1Difference Methods for the Diffusion Equation501
9.2Difference Methods for Poisson Equations517
App. AProofs of Selected Theorems, and other Additional Material535
App. BProofs of Selected Theorems, and other Additional Material542
Index549

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