An Introduction to Numerical Methods and Analysis / Edition 2

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Overview

The objective of the book is for the reader to learn where approximation methods come from, why they work, why they sometimes don't work, and when to use which of many techniques that are available, and to do all this in a style that emphasizes readability and usefulness to the numerical methods novice. Each chapter and each section begins with the basic, elementary material and gradually builds up to more advanced topics.
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Editorial Reviews

From the Publisher
“This is an excellent textbook for a numerical analysis course although it will be difficult to cover all the material in just one course.  Because the text is so complete, it could as well be used for self-study.”  (American Mathematical Society, 1 August 2014)
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Product Details

  • ISBN-13: 9781118367599
  • Publisher: Wiley
  • Publication date: 10/7/2013
  • Edition description: New Edition
  • Edition number: 2
  • Pages: 614
  • Product dimensions: 6.90 (w) x 10.00 (h) x 1.30 (d)

Meet the Author

James F. Epperson, PHD, is Associate Editor of Mathematical Reviews. Dr. Epperson received his PhD from Carnegie Mellon University, and his research interests include the numerical solution of nonlinear evolution equations via finite difference methods, the use of kernel functions to solve evolution equations, and numerical methods in mathematical finance.

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Table of Contents

Preface
Acknowledgments
Ch. 1 Introductory Concepts and Calculus Review 1
1.1 Basic Tools of Calculus 2
1.2 Error, Approximate Equality, and Asymptotic Order Notation 14
1.3 A Primer on Computer Arithmetic 20
1.4 A Word on Computer Languages and Software 28
1.5 Simple Approximations 29
1.6 Application: Approximating the Natural Logarithm 33
Ch. 2 A Survey of Simple Methods and Tools 37
2.1 Homer's Rule and Nested Multiplication 37
2.2 Difference Approximations to the Derivative 40
2.3 Application: Euler's Method for Initial Value Problems 48
2.4 Linear Interpolation 53
2.5 Application: The Trapezoid Rule 60
2.6 Solution of Tridiagonal Linear Systems 69
2.7 Application: Simple Two-Point Boundary Value Problems 75
Ch. 3 Root Finding 80
3.1 The Bisection Method 81
3.2 Newton's Method: Derivation and Examples 87
3.3 How to Stop Newton's Method 93
3.4 Application: Division Using Newton's Method 96
3.5 The Newton Error Formula 100
3.6 Newton's Method: Theory and Convergence 105
3.7 Application: Computation of the Square Root 109
3.8 The Secant Method: Derivation and Example 111
3.9 Fixed-Point Iteration 116
3.10 Special Topics in Root-Finding Methods 126
Ch. 4 Interpolation and Approximation 150
4.1 Lagrange Interpolation 151
4.2 Newton Interpolation and Divided Differences 156
4.3 Interpolation Error 166
4.4 Application: Muller's Method and Inverse Quadratic 170
4.5 Application: More Approximation to the Derivative 174
4.6 Hermite Interpolation 177
4.7 Piecewise Polynomial Interpolation 182
4.8 An Introduction to Splines 189
4.9 Application: Solution of Boundary Value Problems 204
4.10 Least Squares Concepts in Approximation 209
4.11 Advanced Topics in Interpolation Error 230
4.12 Literature and Software Discussion 243
Ch. 5 Numerical Integration 245
5.1 A Review of the Definite Integral 246
5.2 Improving the Trapezoid Rule 248
5.3 Simpson's Rule and Degree of Precision 253
5.4 The Midpoint Rule 265
5.5 Application: Stirling's Formula 268
5.6 Gaussian Quadrature 270
5.7 Extrapolation Methods 281
5.8 Special Topics in Numerical Integration 288
Ch. 6 Numerical Methods for Ordinary Differential Equations 312
6.1 The Initial Value Problem: Background 313
6.2 Euler's Method 318
6.3 Analysis of Euler's Method 322
6.4 Variants of Euler's Method 326
6.5 Single Step Methods: Runge-Kutta 343
6.6 Multistep Methods 350
6.7 Stability Issues 356
6.8 Application to Systems of Equations 363
6.9 Adaptive Solvers 370
6.10 Boundary Value Problems 383
6.11 Literature and Software Discussion 392
Ch. 7 Numerical Methods for the Solution of Systems 394
7.1 Linear Algebra Review 395
7.2 Linear Systems and Gaussian Elimination 397
7.3 Operation Counts 404
7.4 The LU Factorization 406
7.5 Perturbation, Conditioning, and Stability 416
7.6 SPD Matrices and the Cholesky Decomposition 434
7.7 Iterative Methods for Linear Systems: A Brief Survey 437
7.8 Nonlinear Systems: Newton's Method and Related Ideas 446
7.9 Application: Numerical Solution of Nonlinear BVPs 452
7.10 Literature and Software Discussion 454
Ch. 8 Approximate Solution of the Algebraic Eigenvalue Problem 456
8.1 Eigenvalue Review 457
8.2 Reduction to Hessenberg Form 463
8.3 Power Methods 471
8.4 An Overview of the QR Iteration 490
Ch. 9 A Survey of Finite Difference Methods for Partial Differential Equations 500
9.1 Difference Methods for the Diffusion Equation 501
9.2 Difference Methods for Poisson Equations 517
App. A Proofs of Selected Theorems, and other Additional Material 535
App. B Proofs of Selected Theorems, and other Additional Material 542
Index 549
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