Solution Manual to NUMERICAL ANALYSIS FOR ENGINEERS Methods and Applications Second Edition by Ayyub

All 11 Chapters Covered




SOLUTIONS

,Table of Contents
Acknowledgments ............................................................................................................................. iii
Table of Contents ...............................................................................................................................iv
CHAPTER 1. INTRODUCTION ....................................................................................................... 1
1.2 Analytical Versus Numerical Analysis ...................................................................................... 1
1.4 Applications ............................................................................................................................... 1
Computer Programs ......................................................................................................................... 6
CHAPTER 2. MATRICES ................................................................................................................. 9
2.1 Introduction ................................................................................................................................ 9
2.2 Matrix Operations .................................................................................................................... 11
2.3 Vectors ..................................................................................................................................... 14
2.4 Determinants. ........................................................................................................................... 17
2.5 Rank of a Matrix ...................................................................................................................... 18
2.6 Applications ............................................................................................................................. 19
CHAPTER 3. INTRODUCTION TO NUMERICAL METHODS. ................................................. 20
3.1 Introduction .............................................................................................................................. 20
3.2 Accuracy, Precision, and Bias ................................................................................................. 20
3.3 Significant Figures ................................................................................................................... 22
3.4 Analysis of Numerical Errors .................................................................................................. 23
CHAPTER 4. ROOTS OF EQUATIONS......................................................................................... 27
4.1 Introduction .............................................................................................................................. 27
4.2 Eigenvalue Analysis ................................................................................................................ 30
4.3 Direct-Search Method .............................................................................................................. 30
4.4 Bisection Method. .................................................................................................................... 32
4.5 Newton-Raphson Iteration. ...................................................................................................... 35
4.6 Secant Method ......................................................................................................................... 50
4.8 Synthetic Division ................................................................................................................... 55
4.9 Multiple Roots ......................................................................................................................... 70
4.10 Systems of Nonlinear Equations ............................................................................................ 70
CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS. ............................................................ 72
5.2 Gaussian Elimination. .............................................................................................................. 72
5.3 Gauss-Jordan Elimination ........................................................................................................ 74
5.5 LU Decomposition ................................................................................................................... 76
5.6 Iterative Equation-Solving Methods. ....................................................................................... 81
5.6.1 Jacobi Iteration ............................................................................................................................................... 81
5.6.2 Gaussian-Seidel Iteration ................................................................................................................................ 85
5.6.3 Convergence Consideration of the Iterative Methods ..................................................................................... 90
5.7 Use of Determinants ................................................................................................................ 94
5.8 Matrix Inversion ...................................................................................................................... 99
5.9 Applications ........................................................................................................................... 101
Computer Programs ..................................................................................................................... 103
CHAPTER 6. NUMERICAL INTERPOLATION ......................................................................... 105
6.2 Method of Undetermined Coefficients .................................................................................. 105
6.3 Gregory-Newton Interpolation Method ................................................................................. 109
6.4 Finite Difference Interpolation .............................................................................................. 112
6.5 Newton’s Method .................................................................................................................. 114
6.6 Lagrange Polynomials ........................................................................................................... 119
6.7 Interpolation Using Splines ................................................................................................... 124
6.9 Multi-Dimensional Interpolation ........................................................................................... 133
CHAPTER 7. DIFFERENTIATION AND IN @@T
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, 7.1 Numerical Differentiation ...................................................................................................... 135
7.2. Numerical Integration ........................................................................................................... 142
CHAPTER 8. Differential Equations .............................................................................................. 150
8.1 Introduction ............................................................................................................................ 150
8.2 Taylor Series Expansion ........................................................................................................ 150
8.3 Euler’s Method ...................................................................................................................... 154
8.4 Modified Euler’s Method....................................................................................................... 157
8.5 Runge-Kuta Methods ............................................................................................................. 159
8.6 Predictor-Corrector Methods ................................................................................................. 164
8.7 Least-Squares Method ........................................................................................................... 167
8.8 Garlekin Method .................................................................................................................... 170
8.9 Higher-Order Differential Equations ..................................................................................... 172
8.10 Boundary Value Problems ................................................................................................... 172
8.11 Integral Equations ................................................................................................................ 176
CHAPTER 9. Data Description and Treatment .............................................................................. 177
9.2 Classification of Data ............................................................................................................ 177
9.3 Graphical Description of Data ............................................................................................... 177
9.5 Histograms and Frequency Diagrams .................................................................................... 185
9.6 Descriptive Measures............................................................................................................. 187
CHAPTER 10. Curve Fitting and Regression Analysis ................................................................. 190
10.1 Introduction .......................................................................................................................... 190
10.2 Correlation Analysis ............................................................................................................ 190
10.3 Introduction to Regression ................................................................................................... 200
10.4 Principle of Least Squares ................................................................................................... 201
10.5 Reliability of the Regression Equation ................................................................................ 204
10.8 Correlation Versus Regression ............................................................................................ 207
10.9 Application of Bivariate Regression Analysis ..................................................................... 209
10.8 Multiple Regression Analysis .............................................................................................. 213
10.9 Regression Analysis of Nonlinear Models .......................................................................... 220
CHAPTER 11. Numerical Optimization......................................................................................... 238
11.1 Introduction .......................................................................................................................... 238
11.2 The Response Surface Analysis........................................................................................... 238
11.3 Numerical Least Squares ..................................................................................................... 239
11.4 Steepest Descent Method ..................................................................................................... 247




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, CHAPTER 1. INTRODUCTION


1.2 Analytical Versus Numerical Analysis

Problem 1-1.
Solution not provided.


Problem 1-2.
The atwo amethods adiffer aon athe abasis aof atheir arespective aalgorithms. a The aanalytical amethod
ais abased aon aanalytical acalculus awhile athe anumerical amethod ais abased aon afinite adifferences
aarithmetic.
Analytical aapproaches aprovide adirect asolutions aand awill aresult ain aexact asolutions aif athey
aexist. aAnalytical amethods ausually arequire aless atime ato afind aa asolution. a Analytical asolution
aprocedure abecomes aconsiderably amore acomplex awhen aconstraints aare ainvolved. a Numerical
aanalysis, aon athe aother ahand, acan abe aused ato afind asolutions aof amoderately acomplex aproblems,
aand ait ais aquite aeasy ato ainclude aconstraints aon athe aunknowns ain athe asolutions. a However,
anumerical amethods amost aoften arequire aa aconsiderable anumber aof aiterations ain aorder ato afind
aa asolution awith aa areasonable aaccuracy. aThe asolution aprovided aby athe anumerical amethods ais
ausually anot aexact. a Therefore, aerror aanalysis aand aerror aestimations aare arequired.




1.4 Applications

Problem a1-3.
2 4
cos(ax) a = a1 a−xa + xa −.......
2! 4!
For ah a= a0.1
x a= ax0 a + ah a= a0 a+ a0.1 a= a0.1
cos(0.1) a a1.00000000 (one aterm)
(0.1)a2
cos(0.1) a a1 a− = (two aterms)
a0.99500000 a2
(0.1)a2 (0.1)a4
cos(0.1) a 1 a− + = (three aterms)
a 0.99500417
2 24
True avalue a= a0.99500417
The afollowing atable asummarizes athe aresults afor ah a= a0.1 ato a1.0 ain aan aincrement aof a0.1:




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