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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
A solutions manual to accompany An Introduction to Numerical Methods and Analysis, Second Edition
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
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Seitenzahl: 299
Veröffentlichungsjahr: 2014
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Contents
Preface to the Solutions Manual
CHAPTER 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
CHAPTER 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 Exercises:
2.7 APPLICATION: SIMPLE TWO-POINT BOUNDARY VALUE PROBLEMS
CHAPTER 3 ROOT-FINDING
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 Exercises:
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
3.12 VERY HIGH-ORDER METHODS AND THE EFFICIENCY INDEX
CHAPTER 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
CHAPTER 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
CHAPTER 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 MULTI-STEP METHODS
6.7 STABILITY ISSUES
6.8 APPLICATION TO SYSTEMS OF EQUATIONS
6.9 ADAPTIVE SOLVERS
6.10 BOUNDARY VALUE PROBLEMS
CHAPTER 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 BVP'S Exercises:
CHAPTER 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
CHAPTER 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
CHAPTER 10 AN INTRODUCTION TO SPECTRAL METHODS
10.1 SPECTRAL METHODS FOR TWO-POINT BOUNDARY VALUE PROBLEMS
10.2 SPECTRAL METHODS FOR TIME-DEPENDENT PROBLEMS
10.3 CLENSHAW-CURTIS QUADRATURE
Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Epperson, James F., author.
An introduction to numerical methods and analysis/James F. Epperson, Mathematical Reviews. — Second edition.
pages cm
Includes bibliographical references and index.
ISBN 978-1-118-36759-9 (hardback)
1. Numerical analysis. I. Title.
QA297.E568 2013
518—dc23
2013013979
Preface to the Solutions Manual
This manual is written for instructors, not students. It includes worked solutions for many (roughly 75%) of the problems in the text. For the computational exercises I have given the output generated by my program, or sometimes aprogram listing. Most of the programming was done in MATLAB, some in FORTRAN. (The author is well aware that FORTRAN is archaic, but there is a lot of “legacy code” in FORTRAN, and the author believes there is value in learning a new language, even an archaic one.) When the text has a series of exercises that are obviously similar and have similar solutions, then sometimes only one of these problems has a worked solution included. When computational results are asked for a series of similar functions or problems, only a subset of solutions are reported, largely for the sake of brevity. Some exercises that simply ask the student to perform a straight-forward computation are skipped. Exercises that repeat the same computation but with a different method are also often skipped, as are exercises that ask the student to “verify” a straight-forward computation.
Some of the exercises were designed to be open-ended and almost “essay-like.” For these exercises, the only solution typically provided is a short hint or brief outline of the kind of discussion anticipated by the author.
In many exercises the student needs to construct an upper bound on a derivative of some function in order to determine how small a parameter has to be to achieve a desired level of accuracy. For many of the solutions this was done using a computer algebra package and the details are not given.
Students who acquire a copy of this manual in order to obtain worked solutions to homework problems should be aware that none of the solutions are given in enough detail to earn full credit from an instructor.
The author freely admits the potential for error in any of these solutions, especially since many of the exercises were modified after the final version of the text was submitted to the publisher and because the ordering of the exercises was changed from the Revised Edition to the Second Edition. While we tried to make all the appropriate corrections, the possibility of error is still present, and undoubtedly the author's responsibility.
Because much of the manual was constructed by doing “copy-and-paste” from the files for the text, the enumeration of many tables and figures will be different. I have tried to note what the number is in the text, but certainly may have missed some instances.
Suggestions for new exercises and corrections to these solutions are very welcome. Contact the author at [email protected] or [email protected].
Differences from the text The text itself went through a copy-editing process after this manual was completed. As was to be expected, the wording of several problems was slightly changed. None of these changes should affect the problem in terms of what is expected of students; the vast majority of the changes were to replace “previous problem” (a bad habit of mine) with “Problem X.Y” (which I should have done on my own, in the first place). Some puncuation was also changed. The point of adding this note is to explain the textual differences which might be noticed between the text and this manual. If something needs clarification, please contact me at the above email.
