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Book Cover
Author Epperson, James F., author.

Title An introduction to numerical methods and analysis / James F. Epperson.

Published Hoboken, New Jersey : John Wiley & Sons, Inc., [2013]


Location Call No. Status
Edition Second edition.
Physical description xxi, 591 pages ; 26 cm
Notes Previous edition: 2002.
Bibliography Includes bibliographical references and index.
Contents 1.1.Basic Tools of Calculus -- 1.1.1.Taylor's Theorem -- 1.1.2.Mean Value and Extreme Value Theorems -- 1.2.Error, Approximate Equality, and Asymptotic Order Notation -- 1.2.1.Error -- 1.2.2.Notation: Approximate Equality -- 1.2.3.Notation: Asymptotic Order -- 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 -- 2.1.Homer'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 Two-Point Boundary Value Problems -- 3.1.The Bisection Method -- 3.2.Newton's Method: Derivation and Examples -- 3.3.How to Stop Newton's Method --
Contents note continued: 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 Root-finding Methods -- 3.11.1.Extrapolation and Acceleration -- 3.11.2.Variants of Newton's Method -- 3.11.3.The Secant Method: Theory and Convergence -- 3.11.4.Multiple Roots -- 3.11.5.In Search of Fast Global Convergence: Hybrid Algorithms -- 3.12.Very High-order Methods and the Efficiency Index -- 3.13.Literature and Software Discussion -- References -- 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 --
Contents note continued: 4.7.Piecewise Polynomial Interpolation -- 4.8.An Introduction to Splines -- 4.8.1.Definition of the Problem -- 4.8.2.Cubic B-Splines -- 4.9.Application: Solution of Boundary Value Problems -- 4.10.Tension Splines -- 4.11.Least Squares Concepts in Approximation -- 4.11.1.An Introduction to Data Fitting -- 4.11.2.Least Squares Approximation and Orthogonal Polynomials -- 4.12.Advanced Topics in Interpolation Error -- 4.12.1.Stability of Polynomial Interpolation -- 4.12.2.The Runge Example -- 4.12.3.The Chebyshev Nodes -- 4.13.Literature and Software Discussion -- References -- 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 -- 5.8.1.Romberg Integration -- 5.8.2.Quadrature with Non-smooth Integrands --
Contents note continued: 5.8.3.Adaptive Integration -- 5.8.4.Peano Estimates for the Trapezoid Rule -- 5.9.Literature and Software Discussion -- References -- 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 -- 6.4.1.The Residual and Truncation Error -- 6.4.2.Implicit Methods and Predictor-Corrector Schemes -- 6.4.3.Starting Values and Multistep Methods -- 6.4.4.The Midpoint Method and Weak Stability -- 6.5.Single-Step Methods: Runge-Kutta -- 6.6.Multistep Methods -- 6.6.1.The Adams Families -- 6.6.2.The BDF Family -- 6.7.Stability Issues -- 6.7.1.Stability Theory for Multistep Methods -- 6.7.2.Stability Regions -- 6.8.Application to Systems of Equations -- 6.8.1.Implementation Issues and Examples -- 6.8.2.Stiff Equations -- 6.8.3.A-Stability -- 6.9.Adaptive Solvers -- 6.10.Boundary Value Problems -- 6.10.1.Simple Difference Methods -- 6.10.2.Shooting Methods --
Contents note continued: 6.10.3.Finite Element Methods for BVPs -- 6.11.Literature and Software Discussion -- References -- 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 -- 7.5.1.Vector and Matrix Norms -- 7.5.2.The Condition Number and Perturbations -- 7.5.3.Estimating the Condition Number -- 7.5.4.Iterative Refinement -- 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 -- 7.8.1.Newton's Method -- 7.8.2.Fixed-Point Methods -- 7.9.Application: Numerical Solution of Nonlinear Boundary Value Problems -- 7.10.Literature and Software Discussion -- References -- 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, Part II --
Contents note continued: 8.6.Literature and Software Discussion -- References -- 9.1.Difference Methods for the Diffusion Equation -- 9.1.1.The Basic Problem -- 9.1.2.The Explicit Method and Stability -- 9.1.3.Implicit Methods and the Crank-Nicolson Method -- 9.2.Finite Element Methods for the Diffusion Equation -- 9.3.Difference Methods for Poisson Equations -- 9.3.1.Discretization -- 9.3.2.Banded Cholesky Solvers -- 9.3.3.Iteration and the Method of Conjugate Gradients -- 9.4.Literature and Software Discussion -- References -- 10.1.Spectral Methods for Two-Point Boundary Value Problems -- 10.2.Spectral Methods for Time-Dependent Problems -- 10.3.Clenshaw-Curtis Quadrature -- 10.4.Literature and Software Discussion -- References -- 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.
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"--Provided by publisher.
Subject Numerical analysis.
ISBN 9781118367599 (hardback)
1118367596 (hardback)