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9780470049631

An Introduction to Numerical Methods and Analysis

by
  • ISBN13:

    9780470049631

  • ISBN10:

    0470049634

  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2007-09-17
  • Publisher: Wiley-Interscience
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List Price: $159.00

Summary

The objective of this 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 way 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 as "starter exercises", to challenging derivations and minor proofs, to programming exercises.

Author Biography

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.

Table of Contents

Preface
Introductory Concepts and Calculus Review
Basic Tools of Calculus
Taylora??s Theorem
Mean Value and Extreme Value Theorems
Error, Approximate Equality, and Asymptotic Order Notation
Error
Notation: Approximate Equality
Notation: Asymptotic Order
A Primer on Computer Arithmetic
A Word on Computer Languages and Software
Simple Approximations
Application: Approximating the Natural Logarithm
References
A Survey of Simple Methods and Tools
Horner's Rule and Nested Multiplication
Difference Approximations to the Derivative
Application: Euler's Method for Initial Value Problems
Linear Interpolation
Application - The Trapezoid Rule
Solution of Tridiagonal Linear Systems
Application: Simple Two-Point Boundary Value Problems
Root-Finding
The Bisection Method
Newtona??s Method: Derivation and Examples
How to Stop Newton's Method
Application: Division Using Newton's Method
The Newton Error Formula
Newtona??s Method: Theory and Convergence
Application: Computation of the Square Root
The Secant Method: Derivation and Examples
Fixed Point Iteration
Special Topics in Root-finding Methods
Extrapolation and Acceleration
Variants of Newtona??s Method
The Secant Method: Theory and Convergence
Multiple Roots
In Search of Fast Global Convergence: Hybrid Algorithms
Literature and Software Discussion 56
References
Interpolation and Approximation
Lagrange Interpolation
Interpolation and Divided Differences
Interpolation Error
Application: Muller's Method and Inverse Quadratic Interpolation
Application: More Approximations to the Derivative
Hermite Interpolation
Piecewise Polynomial Interpolation
An Introduction to Splines
Definition of the Problem
Cubic B-Splines
Application: Solution of Boundary Value Problems
Least Squares Concepts in Approximation
An Introduction to Data Fitting
Least Squares Approximation and Orthogonal Polynomials
Advanced Topics in Interpolation Error
Stability of Polynomial Interpolation
The Runge Example
The Chebyshev nodes
Literature and Software Discussion
References
Numerical Integration
A Review of the Definite Integral
Improving the Trapezoid Rule
Simpson's Rule and Degree of Precision
The Midpoint Rule
Application: Stirlinga??s Formula
Gaussian Quadrature
Extrapolation Methods
Special Topics in Numerical Integration
Romberg Integration
Quadrature with Non-Smooth Integrands
Adaptive Integration
Peano Estimates for the Trapezoid Rule
Literature and Software Discussion
References
Numerical Methods for Ordinary Differential Equations
The Initial Value Problem - Background
Euler's Method
Analysis of Euler's Method
Variants of Euler's Method
The Residual and Truncation Error
Implicit Methods and Predictor-Corrector Schemes
Starting Values and Multistep Methods
The Midpoint Method and Weak Stability
Single Step Methods? Runge-Kutta
Multi-step Methods
The Adams Families
The BDF Family
Stability Issues
Stability Theory for Multistep Methods
Stability Regions
Application to Systems of Equations
Implementation Issues and Examples
Stiff Equations
A-Stability
Adaptive Solvers
Boundary Value Problems
Simple Difference Methods
Shooting Methods
Literature and Software Discussion
References
Numerical Methods for the Solution of Systems
Table of Contents provided by Publisher. All Rights Reserved.

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