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
SeSE G
iesimiciR
sm sioA
iic sloaltT
aiotinI
oOn N ......................................................... 135
iv
, 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
@@
SeSiesim
smiciii csiosloaltaiotinon
v
, CHAPTER 1. INTRODUCTION
1.2 Analytical Versus Numerical Analysis
Problem 1-1.
Solution not provided.
Problem 1-2.
The stwo smethods sdiffer son sthe sbasis sof stheir srespective salgorithms. s The sanalytical smethod sis
sbased son sanalytical scalculus swhile sthe snumerical smethod sis sbased son sfinite sdifferences
sarithmetic.
Analytical sapproaches sprovide sdirect ssolutions sand swill sresult sin sexact ssolutions sif sthey sexist.
sAnalytical smethods susually srequire sless stime sto sfind sa ssolution. s Analytical ssolution
sprocedure sbecomes sconsiderably smore scomplex swhen sconstraints sare sinvolved. s Numerical
sanalysis, son sthe sother shand, scan sbe sused sto sfind ssolutions sof smoderately scomplex sproblems,
sand sit sis squite seasy sto sinclude sconstraints son sthe sunknowns sin sthe ssolutions. s However,
snumerical smethods smost soften srequire sa sconsiderable snumber sof siterations sin sorder sto sfind sa
ssolution swith sa sreasonable saccuracy. sThe ssolution sprovided sby sthe snumerical smethods sis
susually snot sexact. s Therefore, serror sanalysis sand serror sestimations sare srequired.
1.4 Applications
Problem s1-3.
2 4
cos(sx) s = s1 s−xs + xs −.......
2! 4!
For sh s= s0.1
x s= sx0 s + sh s= s0 s+ s0.1 s= s0.1
cos(0.1) s s1.00000000 (one sterm)
(0.1)s2
cos(0.1) s s1 s− = (two sterms)
s0.99500000 s2
(0.1)s2 (0.1)s4
cos(0.1) s 1 s− + = (three sterms)
s0.99500417
2 24
True svalue s= s0.99500417
The sfollowing stable ssummarizes sthe sresults sfor sh s= s0.1 sto s1.0 sin san sincrement sof s0.1:
@@
SeSiesimiciiicsiosloaltaiotinon
sm
1
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
SeSE G
iesimiciR
sm sioA
iic sloaltT
aiotinI
oOn N ......................................................... 135
iv
, 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
@@
SeSiesim
smiciii csiosloaltaiotinon
v
, CHAPTER 1. INTRODUCTION
1.2 Analytical Versus Numerical Analysis
Problem 1-1.
Solution not provided.
Problem 1-2.
The stwo smethods sdiffer son sthe sbasis sof stheir srespective salgorithms. s The sanalytical smethod sis
sbased son sanalytical scalculus swhile sthe snumerical smethod sis sbased son sfinite sdifferences
sarithmetic.
Analytical sapproaches sprovide sdirect ssolutions sand swill sresult sin sexact ssolutions sif sthey sexist.
sAnalytical smethods susually srequire sless stime sto sfind sa ssolution. s Analytical ssolution
sprocedure sbecomes sconsiderably smore scomplex swhen sconstraints sare sinvolved. s Numerical
sanalysis, son sthe sother shand, scan sbe sused sto sfind ssolutions sof smoderately scomplex sproblems,
sand sit sis squite seasy sto sinclude sconstraints son sthe sunknowns sin sthe ssolutions. s However,
snumerical smethods smost soften srequire sa sconsiderable snumber sof siterations sin sorder sto sfind sa
ssolution swith sa sreasonable saccuracy. sThe ssolution sprovided sby sthe snumerical smethods sis
susually snot sexact. s Therefore, serror sanalysis sand serror sestimations sare srequired.
1.4 Applications
Problem s1-3.
2 4
cos(sx) s = s1 s−xs + xs −.......
2! 4!
For sh s= s0.1
x s= sx0 s + sh s= s0 s+ s0.1 s= s0.1
cos(0.1) s s1.00000000 (one sterm)
(0.1)s2
cos(0.1) s s1 s− = (two sterms)
s0.99500000 s2
(0.1)s2 (0.1)s4
cos(0.1) s 1 s− + = (three sterms)
s0.99500417
2 24
True svalue s= s0.99500417
The sfollowing stable ssummarizes sthe sresults sfor sh s= s0.1 sto s1.0 sin san sincrement sof s0.1:
@@
SeSiesimiciiicsiosloaltaiotinon
sm
1
