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SOLUTIONS MANUAL for An Introduction to Numerical
Methods and Analysis 2nd Edition by James F. Epperson
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,
,Solutions Manual to Accompany
AN INTRODUCTION TO
NUMERICAL METHODS
AND ANALYSIS
Second Edition
JAMES F. EPPERSON
Mathematical Reviews
Downloaded from Scholarfriends.com
,Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or
by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as
permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior
written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to
the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax
(978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should
be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ
07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in
preparing this book, they make no representation or warranties with respect to the accuracy or
completeness of the contents of this book and specifically disclaim any implied warranties of
merchantability or fitness for a particular purpose. No warranty may be created or extended by sales
representatives or written sales materials. The advice and strategies contained herein may not be suitable
for your situation. You should consult with a professional where appropriate. Neither the publisher nor
author shall be liable for any loss of profit or any other commercial damages, including but not limited
to special, incidental, consequential, or other damages.
For general information on our other products and services please contact our Customer Care
Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or
fax (317) 572-4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print,
however, may not be available in electronic formats. For more information about Wiley products, visit
our web site at www.wiley.com.
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
Downloaded from Scholarfriends.com
10 9 8 7 6 5 4 3 2 1
,CONTENTS
1 Introductory Concepts and Calculus Review 1
1.1 Basic Tools of Calculus 1
1.2 Error, Approximate Equality, and Asymptotic Order Notation 12
1.3 A Primer on Computer Arithmetic 15
1.4 A Word on Computer Languages and Software 19
1.5 Simple Approximations 19
1.6 Application: Approximating the Natural Logarithm 22
1.7 A Brief History of Computing 25
2 A Survey of Simple Methods and Tools 27
2.1 Horner’s Rule and Nested Multiplication 27
2.2 Difference Approximations to the Derivative 30
2.3 Application: Euler’s Method for Initial Value Problems 40
2.4 Linear Interpolation 44
2.5 Application — The Trapezoid Rule 48
2.6 Solution of Tridiagonal Linear Systems 56
Downloaded from Scholarfriends.com
2.7 Application: Simple Two-Point Boundary Value Problems 61
v
,vi CONTENTS
3 Root-Finding 65
3.1 The Bisection Method 65
3.2 Newton’s Method: Derivation and Examples 69
3.3 How to Stop Newton’s Method 73
3.4 Application: Division Using Newton’s Method 77
3.5 The Newton Error Formula 81
3.6 Newton’s Method: Theory and Convergence 84
3.7 Application: Computation of the Square Root 88
3.8 The Secant Method: Derivation and Examples 92
3.9 Fixed Point Iteration 96
3.10 Roots of Polynomials (Part 1) 99
3.11 Special Topics in Root-finding Methods 102
3.12 Very High-order Methods and the Efficiency Index 114
4 Interpolation and Approximation 117
4.1 Lagrange Interpolation 117
4.2 Newton Interpolation and Divided Differences 120
4.3 Interpolation Error 132
4.4 Application: Muller’s Method and Inverse Quadratic
Interpolation 139
4.5 Application: More Approximations to the Derivative 141
4.6 Hermite Interpolation 142
4.7 Piecewise Polynomial Interpolation 145
4.8 An Introduction to Splines 149
4.9 Application: Solution of Boundary Value Problems 156
4.10 Tension Splines 159
4.11 Least Squares Concepts in Approximation 160
4.12 Advanced Topics in Interpolation Error 166
5 Numerical Integration 171
5.1 A Review of the Definite Integral 171
5.2 Improving the Trapezoid Rule 173
5.3 Simpson’s Rule and Degree of Precision 177
5.4 The Midpoint Rule 187
5.5 Application: Stirling’s Formula 190
5.6 Gaussian Quadrature 192
5.7 Extrapolation Methods 199
Downloaded from Scholarfriends.com
5.8 Special Topics in Numerical Integration 203
, CONTENTS vii
6 Numerical Methods for Ordinary Differential Equations 211
6.1 The Initial Value Problem — Background 211
6.2 Euler’s Method 213
6.3 Analysis of Euler’s Method 216
6.4 Variants of Euler’s Method 217
6.5 Single Step Methods — Runge-Kutta 225
6.6 Multi-step Methods 228
6.7 Stability Issues 234
6.8 Application to Systems of Equations 235
6.9 Adaptive Solvers 240
6.10 Boundary Value Problems 243
7 Numerical Methods for the Solution of Systems of Equations 247
7.1 Linear Algebra Review 247
7.2 Linear Systems and Gaussian Elimination 248
7.3 Operation Counts 254
7.4 The LU Factorization 256
7.5 Perturbation, Conditioning and Stability 262
7.6 SP D Matrices and the Cholesky Decomposition 269
7.7 Iterative Methods for Linear Systems – A Brief Survey 271
7.8 Nonlinear Systems: Newton’s Method and Related Ideas 273
7.9 Application: Numerical Solution of Nonlinear BVP’s 275
8 Approximate Solution of the Algebraic Eigenvalue Problem 277
8.1 Eigenvalue Review 277
8.2 Reduction to Hessenberg Form 280
8.3 Power Methods 281
8.4 An Overview of the QR Iteration 284
8.5 Application: Roots of Polynomials, II 288
9 A Survey of Numerical Methods
for Partial Differential Equations 289
9.1 Difference Methods for the Diffusion Equation 289
9.2 Finite Element Methods for the Diffusion Equation 293
Downloaded from Scholarfriends.com
9.3 Difference Methods for Poisson Equations 294
,viii CONTENTS
10 An Introduction to Spectral Methods 299
10.1 Spectral Methods for Two-Point Boundary Value Problems 299
10.2 Spectral Methods for Time-Dependent Problems 301
10.3 Clenshaw-Curtis Quadrature 303
Downloaded from Scholarfriends.com
,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 a program 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
Downloaded from Scholarfriends.com
ix
, x
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 or .
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.
Downloaded from Scholarfriends.com
Scholarfriends
SOLUTIONS MANUAL for An Introduction to Numerical
Methods and Analysis 2nd Edition by James F. Epperson
https://scholarfriends.com
Downloaded by: | https://scholarfriends.com/singlePaper/638933/solutions-manual-for-an-
introduction-to-numerical-methods-and-analysis-2nd-edition-by-james-f-epper
Distribution of this document is illegal.
,
,Solutions Manual to Accompany
AN INTRODUCTION TO
NUMERICAL METHODS
AND ANALYSIS
Second Edition
JAMES F. EPPERSON
Mathematical Reviews
Downloaded from Scholarfriends.com
,Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or
by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as
permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior
written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to
the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax
(978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should
be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ
07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in
preparing this book, they make no representation or warranties with respect to the accuracy or
completeness of the contents of this book and specifically disclaim any implied warranties of
merchantability or fitness for a particular purpose. No warranty may be created or extended by sales
representatives or written sales materials. The advice and strategies contained herein may not be suitable
for your situation. You should consult with a professional where appropriate. Neither the publisher nor
author shall be liable for any loss of profit or any other commercial damages, including but not limited
to special, incidental, consequential, or other damages.
For general information on our other products and services please contact our Customer Care
Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or
fax (317) 572-4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print,
however, may not be available in electronic formats. For more information about Wiley products, visit
our web site at www.wiley.com.
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
Downloaded from Scholarfriends.com
10 9 8 7 6 5 4 3 2 1
,CONTENTS
1 Introductory Concepts and Calculus Review 1
1.1 Basic Tools of Calculus 1
1.2 Error, Approximate Equality, and Asymptotic Order Notation 12
1.3 A Primer on Computer Arithmetic 15
1.4 A Word on Computer Languages and Software 19
1.5 Simple Approximations 19
1.6 Application: Approximating the Natural Logarithm 22
1.7 A Brief History of Computing 25
2 A Survey of Simple Methods and Tools 27
2.1 Horner’s Rule and Nested Multiplication 27
2.2 Difference Approximations to the Derivative 30
2.3 Application: Euler’s Method for Initial Value Problems 40
2.4 Linear Interpolation 44
2.5 Application — The Trapezoid Rule 48
2.6 Solution of Tridiagonal Linear Systems 56
Downloaded from Scholarfriends.com
2.7 Application: Simple Two-Point Boundary Value Problems 61
v
,vi CONTENTS
3 Root-Finding 65
3.1 The Bisection Method 65
3.2 Newton’s Method: Derivation and Examples 69
3.3 How to Stop Newton’s Method 73
3.4 Application: Division Using Newton’s Method 77
3.5 The Newton Error Formula 81
3.6 Newton’s Method: Theory and Convergence 84
3.7 Application: Computation of the Square Root 88
3.8 The Secant Method: Derivation and Examples 92
3.9 Fixed Point Iteration 96
3.10 Roots of Polynomials (Part 1) 99
3.11 Special Topics in Root-finding Methods 102
3.12 Very High-order Methods and the Efficiency Index 114
4 Interpolation and Approximation 117
4.1 Lagrange Interpolation 117
4.2 Newton Interpolation and Divided Differences 120
4.3 Interpolation Error 132
4.4 Application: Muller’s Method and Inverse Quadratic
Interpolation 139
4.5 Application: More Approximations to the Derivative 141
4.6 Hermite Interpolation 142
4.7 Piecewise Polynomial Interpolation 145
4.8 An Introduction to Splines 149
4.9 Application: Solution of Boundary Value Problems 156
4.10 Tension Splines 159
4.11 Least Squares Concepts in Approximation 160
4.12 Advanced Topics in Interpolation Error 166
5 Numerical Integration 171
5.1 A Review of the Definite Integral 171
5.2 Improving the Trapezoid Rule 173
5.3 Simpson’s Rule and Degree of Precision 177
5.4 The Midpoint Rule 187
5.5 Application: Stirling’s Formula 190
5.6 Gaussian Quadrature 192
5.7 Extrapolation Methods 199
Downloaded from Scholarfriends.com
5.8 Special Topics in Numerical Integration 203
, CONTENTS vii
6 Numerical Methods for Ordinary Differential Equations 211
6.1 The Initial Value Problem — Background 211
6.2 Euler’s Method 213
6.3 Analysis of Euler’s Method 216
6.4 Variants of Euler’s Method 217
6.5 Single Step Methods — Runge-Kutta 225
6.6 Multi-step Methods 228
6.7 Stability Issues 234
6.8 Application to Systems of Equations 235
6.9 Adaptive Solvers 240
6.10 Boundary Value Problems 243
7 Numerical Methods for the Solution of Systems of Equations 247
7.1 Linear Algebra Review 247
7.2 Linear Systems and Gaussian Elimination 248
7.3 Operation Counts 254
7.4 The LU Factorization 256
7.5 Perturbation, Conditioning and Stability 262
7.6 SP D Matrices and the Cholesky Decomposition 269
7.7 Iterative Methods for Linear Systems – A Brief Survey 271
7.8 Nonlinear Systems: Newton’s Method and Related Ideas 273
7.9 Application: Numerical Solution of Nonlinear BVP’s 275
8 Approximate Solution of the Algebraic Eigenvalue Problem 277
8.1 Eigenvalue Review 277
8.2 Reduction to Hessenberg Form 280
8.3 Power Methods 281
8.4 An Overview of the QR Iteration 284
8.5 Application: Roots of Polynomials, II 288
9 A Survey of Numerical Methods
for Partial Differential Equations 289
9.1 Difference Methods for the Diffusion Equation 289
9.2 Finite Element Methods for the Diffusion Equation 293
Downloaded from Scholarfriends.com
9.3 Difference Methods for Poisson Equations 294
,viii CONTENTS
10 An Introduction to Spectral Methods 299
10.1 Spectral Methods for Two-Point Boundary Value Problems 299
10.2 Spectral Methods for Time-Dependent Problems 301
10.3 Clenshaw-Curtis Quadrature 303
Downloaded from Scholarfriends.com
,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 a program 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
Downloaded from Scholarfriends.com
ix
, x
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 or .
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.
Downloaded from Scholarfriends.com
