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Introduction to numerical methods and analysis / James F. Epperson.

By: Material type: TextTextPublication details: New Jersey : John Wiley, c2013.Edition: 2nd edDescription: xxi, 591 p. : ill. ; 26 cmISBN:
  • 9781118367599 (hardback)
Subject(s): Additional physical formats: Online version:: Introduction to numerical methods and analysisDDC classification:
  • 518 23 Ep64
Contents:
Preface-- 1. Introductory Concepts and Calculus Review-- 1.1 Basic Tools of Calculus-- 1.2 Error, Approximate Equality, and Asymptotic Order Notation -- 1.3 A Primer on Computer Arithmetic-- 1.4 A Word on Computer Languages and Software-- 1.5 Simple Approximations-- 1.6 Application: Approximating the Natural Logarithm-- 1.7 A Brief History of Computing-- 1.8 Literature Review-- References-- 2 A Survey of Simple Methods and Tools-- 2.1 Horner's Rule and Nested Multiplication-- 2.2 Difference Approximations to the Derivative-- 2.3 Application: Euler's Method for Initial Value Problems-- 2.4 Linear Interpolation-- 2.5 Application--The Trapezoid Rule-- 2.6 Solution of Tridiagonal Linear Systems-- 2.7 Application: Simple TwoPoint Boundary Value Problems-- 3 RootFinding-- 3.1 The Bisection Method-- 3.2 Newton's Method: Derivation and Examples-- 3.3 How to Stop Newton's Method-- 3.4 Application: Division Using Newton's Method-- 3.5 The Newton Error Formula-- 3.6 Newton's Method: Theory and Convergence-- 3.7 Application: Computation of the Square Root-- 3.8 The Secant Method: Derivation and Examples-- 3.9 Fixed Point Iteration-- 3.10 Roots of Polynomials (Part 1)-- 3.11 Special Topics in Rootfinding Methods-- 3.12 Very Highorder Methods and the Efficiency Index-- 3.13 Literature and Software Discussion-- References-- 4 Interpolation and Approximation-- 4.1 Lagrange Interpolation-- 4.2 Newton Interpolation and Divided Differences-- 4.3 Interpolation Error-- 4.4 Application: Muller's Method and Inverse Quadratic Interpolation-- 4.5 Application: More Approximations to the Derivative-- 4.6 Hermite Interpolation-- 4.7 Piecewise Polynomial Interpolation-- 4.8 An Introduction to Splines-- 4.9 Application: Solution of Boundary Value Problems-- 4.10 Tension Splines-- 4.11 Least Squares Concepts in Approximation-- 4.12 Advanced Topics in Interpolation Error-- 4.13 Literature and Software Discussion-- References-- 5 Numerical Integration-- 5.1 A Review of the Definite Integral-- 5.2 Improving the Trapezoid Rule-- 5.3 Simpson's Rule and Degree of Precision-- 5.4 The Midpoint Rule-- 5.5 Application: Stirling's Formula-- 5.6 Gaussian Quadrature-- 5.7 Extrapolation Methods-- 5.8 Special Topics in Numerical Integration-- 5.9 Literature and Software Discussion-- References 6 Numerical Methods for Ordinary Differential Equations-- 6.1 The Initial Value Problem--Background-- 6.2 Euler's Method-- 6.3 Analysis of Euler's Method-- 6.4 Variants of Euler's Method-- 6.5 Single Step Methods--Runge--Kutta-- 6.6 Multistep Methods-- 6.7 Stability Issues-- 6.8 Application to Systems of Equations-- 6.9 Adaptive Solvers-- 6.10 Boundary Value Problems-- 6.11 Literature and Software Discussion-- References 7 Numerical Methods for the Solution of Systems of Equations-- 7.1 Linear Algebra Review-- 7.2 Linear Systems and Gaussian Elimination-- 7.3 Operation Counts-- 7.4 The LU Factorization-- 7.5 Perturbation, Conditioning, and Stability-- 7.6 SPD Matrices and the Cholesky Decomposition-- 7.7 Iterative Methods for Linear Systems--A Brief Survey-- 7.8 Nonlinear Systems: Newton's Method and Related Ideas- 7.9 Application: Numerical Solution of Nonlinear Boundary Value Problems-- 7.10 Literature and Software Discussion 487 References-- References-- 8 Approximate Solution of the Algebraic Eigenvalue Problem-- 8.1 Eigenvalue Review-- 8.2 Reduction to Hessenberg Form-- 8.3 Power Methods-- 8.4 An Overview of the QR Iteration-- 8.5 Application: Roots of Polynomials, II-- 8.6 Literature and Software Discussion-- References-- 9 A Survey of Numerical Methods for Partial Differential Equations-- 9.1 Difference Methods for the Diffusion Equation-- 9.2 Finite Element Methods for the Diffusion Equation-- 9.3 Difference Methods for Poisson Equations-- 9.4 Literature and Software Discussion-- References-- 10 An Introduction to Spectral Methods-- 10.1 Spectral Methods for TwoPoint Boundary Value Problems -- 10.2 Spectral Methods for TimeDependent Problems-- 10.3 ClenshawCurtis Quadrature-- 10.4 Literature and Software Discussion-- References-- Appendix A: Proofs of Selected Theorems, and Other Additional Material-- A.1 Proofs of the Interpolation Error Theorems-- A.2 Proof of the Stability Result for ODEs-- A.3 Stiff Systems of Differential Equations and Eigenvalues-- A.4 The Matrix Perturbation Theorem-- Index--
Summary: "The objective of this book is for readers to learn where approximation methods come from, why they work, why they sometimes don't work, and when to use which of the many techniques that are available, and to do all this in an environment 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. The text begins with a review of the important calculus results, and why and where these ideas play an important role throughout the book. Some of the concepts required for the study of computational mathematics are introduced, and simple approximations using Taylor's Theorem are treated in some depth. The exposition is intended to be lively and "student friendly". Exercises run the gamut from simple hand computations that might be characterized are "starter exercises", to challenging derivations and minor proofs, to programming exercises. Eleven new exercises have been added throughout including: Basins of Attraction; Roots of Polynomials I; Radial Basis Function Interpolation; Tension Splines; An Introduction to Galerkin/Finite Element Ideas for BVPs; Broyden's Method; Roots of Polynomials, II; Spectral/collocation methods for PDEs; Algebraic Multigrid Method; Trigonometric interpolation/Fourier analysis; and Monte Carlo methods. Various sections have been revised to reflect recent trends and updates in the field"--
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Includes bibliographical references and index.

Preface--
1. Introductory Concepts and Calculus Review--
1.1 Basic Tools of Calculus--
1.2 Error, Approximate Equality, and Asymptotic Order Notation --
1.3 A Primer on Computer Arithmetic--
1.4 A Word on Computer Languages and Software--
1.5 Simple Approximations--
1.6 Application: Approximating the Natural Logarithm--
1.7 A Brief History of Computing--
1.8 Literature Review--
References--

2 A Survey of Simple Methods and Tools--
2.1 Horner's Rule and Nested Multiplication--
2.2 Difference Approximations to the Derivative--
2.3 Application: Euler's Method for Initial Value Problems--
2.4 Linear Interpolation--
2.5 Application--The Trapezoid Rule--
2.6 Solution of Tridiagonal Linear Systems--
2.7 Application: Simple TwoPoint Boundary Value Problems--

3 RootFinding--
3.1 The Bisection Method--
3.2 Newton's Method: Derivation and Examples--
3.3 How to Stop Newton's Method--
3.4 Application: Division Using Newton's Method--
3.5 The Newton Error Formula--
3.6 Newton's Method: Theory and Convergence--
3.7 Application: Computation of the Square Root--
3.8 The Secant Method: Derivation and Examples--
3.9 Fixed Point Iteration--
3.10 Roots of Polynomials (Part 1)--
3.11 Special Topics in Rootfinding Methods--
3.12 Very Highorder Methods and the Efficiency Index--
3.13 Literature and Software Discussion--
References--

4 Interpolation and Approximation--
4.1 Lagrange Interpolation--
4.2 Newton Interpolation and Divided Differences--
4.3 Interpolation Error--
4.4 Application: Muller's Method and Inverse Quadratic Interpolation--
4.5 Application: More Approximations to the Derivative--
4.6 Hermite Interpolation--
4.7 Piecewise Polynomial Interpolation--
4.8 An Introduction to Splines--
4.9 Application: Solution of Boundary Value Problems--
4.10 Tension Splines--
4.11 Least Squares Concepts in Approximation--
4.12 Advanced Topics in Interpolation Error--
4.13 Literature and Software Discussion--
References--

5 Numerical Integration--
5.1 A Review of the Definite Integral--
5.2 Improving the Trapezoid Rule--
5.3 Simpson's Rule and Degree of Precision--
5.4 The Midpoint Rule--
5.5 Application: Stirling's Formula--
5.6 Gaussian Quadrature--
5.7 Extrapolation Methods--
5.8 Special Topics in Numerical Integration--
5.9 Literature and Software Discussion--
References

6 Numerical Methods for Ordinary Differential Equations--
6.1 The Initial Value Problem--Background--
6.2 Euler's Method--
6.3 Analysis of Euler's Method--
6.4 Variants of Euler's Method--
6.5 Single Step Methods--Runge--Kutta--
6.6 Multistep Methods--
6.7 Stability Issues--
6.8 Application to Systems of Equations--
6.9 Adaptive Solvers--
6.10 Boundary Value Problems--
6.11 Literature and Software Discussion--
References

7 Numerical Methods for the Solution of Systems of Equations--
7.1 Linear Algebra Review--
7.2 Linear Systems and Gaussian Elimination--
7.3 Operation Counts--
7.4 The LU Factorization--
7.5 Perturbation, Conditioning, and Stability--
7.6 SPD Matrices and the Cholesky Decomposition--
7.7 Iterative Methods for Linear Systems--A Brief Survey--
7.8 Nonlinear Systems: Newton's Method and Related Ideas-
7.9 Application: Numerical Solution of Nonlinear Boundary Value Problems--
7.10 Literature and Software Discussion 487 References--
References--

8 Approximate Solution of the Algebraic Eigenvalue Problem-- 8.1 Eigenvalue Review--
8.2 Reduction to Hessenberg Form--
8.3 Power Methods--
8.4 An Overview of the QR Iteration--
8.5 Application: Roots of Polynomials, II--
8.6 Literature and Software Discussion--
References--

9 A Survey of Numerical Methods for Partial Differential Equations--
9.1 Difference Methods for the Diffusion Equation--
9.2 Finite Element Methods for the Diffusion Equation--
9.3 Difference Methods for Poisson Equations--
9.4 Literature and Software Discussion--
References--

10 An Introduction to Spectral Methods--
10.1 Spectral Methods for TwoPoint Boundary Value Problems --
10.2 Spectral Methods for TimeDependent Problems--
10.3 ClenshawCurtis Quadrature--
10.4 Literature and Software Discussion--
References--

Appendix A: Proofs of Selected Theorems, and Other Additional Material--
A.1 Proofs of the Interpolation Error Theorems--
A.2 Proof of the Stability Result for ODEs--
A.3 Stiff Systems of Differential Equations and Eigenvalues-- A.4 The Matrix Perturbation Theorem--
Index--

"The objective of this book is for readers to learn where approximation methods come from, why they work, why they sometimes don't work, and when to use which of the many techniques that are available, and to do all this in an environment 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. The text begins with a review of the important calculus results, and why and where these ideas play an important role throughout the book. Some of the concepts required for the study of computational mathematics are introduced, and simple approximations using Taylor's Theorem are treated in some depth. The exposition is intended to be lively and "student friendly". Exercises run the gamut from simple hand computations that might be characterized are "starter exercises", to challenging derivations and minor proofs, to programming exercises. Eleven new exercises have been added throughout including: Basins of Attraction; Roots of Polynomials I; Radial Basis Function Interpolation; Tension Splines; An Introduction to Galerkin/Finite Element Ideas for BVPs; Broyden's Method; Roots of Polynomials, II; Spectral/collocation methods for PDEs; Algebraic Multigrid Method; Trigonometric interpolation/Fourier analysis; and Monte Carlo methods. Various sections have been revised to reflect recent trends and updates in the field"--

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