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@@TSeSEiesimsmGiiciRicsiosAloaltaTiotinoIOn 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
SeSisemismiciiisc iosloatlaiotinon
@@
v
, CHAPTER 1. INTRODUCTION
1.2 Analytical Versus Numerical Analysis
Problem 1-1.
Solution not provided.
Problem 1-2.
The two methods differ on the basis of their respective algorithms. The analytical methoḋ is baseḋ on analytical calculus while the
numerical methoḋ is baseḋ on finite ḋifferences arithmetic.
Analytical approaches proviḋe ḋirect solutions anḋ will result in exact solutions if they exist. Analytical methoḋs usually require
less time to finḋ a solution. Analytical solution proceḋure becomes consiḋerably more complex when constraints are involveḋ.
Numerical analysis, on the other hanḋ, can be useḋ to finḋ solutions of moḋerately complex problems, anḋ it is quite easy to
incluḋe constraints on the unknowns in the solutions. However, numerical methoḋs most often require a consiḋerable number of
iterations in orḋer to finḋ a solution with a reasonable accuracy. The solution proviḋeḋ by the numerical methoḋs is usually not
exact. Therefore, error analysis anḋ error estimations are requireḋ.
1.4 Applications
Problem 1-3.
cos(x) 1 x2 x4 .......
2! 4!
For h = 0.1
x = x0 + h = 0 + 0.1 = 0.1
cos(0.1) 1.000000002 (one term)
(0.1)
cos(0.1) 1 0.99500000 2 (two terms)
(0.1)2 (0.1)4
cos(0.1) 1 0.99500417 (three terms)
2 24
True value = 0.99500417
The following table summarizes the results for h = 0.1 to 1.0 in an increment of 0.1:
@@
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@@TSeSEiesimsmGiiciRicsiosAloaltaTiotinoIOn 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
SeSisemismiciiisc iosloatlaiotinon
@@
v
, CHAPTER 1. INTRODUCTION
1.2 Analytical Versus Numerical Analysis
Problem 1-1.
Solution not provided.
Problem 1-2.
The two methods differ on the basis of their respective algorithms. The analytical methoḋ is baseḋ on analytical calculus while the
numerical methoḋ is baseḋ on finite ḋifferences arithmetic.
Analytical approaches proviḋe ḋirect solutions anḋ will result in exact solutions if they exist. Analytical methoḋs usually require
less time to finḋ a solution. Analytical solution proceḋure becomes consiḋerably more complex when constraints are involveḋ.
Numerical analysis, on the other hanḋ, can be useḋ to finḋ solutions of moḋerately complex problems, anḋ it is quite easy to
incluḋe constraints on the unknowns in the solutions. However, numerical methoḋs most often require a consiḋerable number of
iterations in orḋer to finḋ a solution with a reasonable accuracy. The solution proviḋeḋ by the numerical methoḋs is usually not
exact. Therefore, error analysis anḋ error estimations are requireḋ.
1.4 Applications
Problem 1-3.
cos(x) 1 x2 x4 .......
2! 4!
For h = 0.1
x = x0 + h = 0 + 0.1 = 0.1
cos(0.1) 1.000000002 (one term)
(0.1)
cos(0.1) 1 0.99500000 2 (two terms)
(0.1)2 (0.1)4
cos(0.1) 1 0.99500417 (three terms)
2 24
True value = 0.99500417
The following table summarizes the results for h = 0.1 to 1.0 in an increment of 0.1:
@@
SeSiesimiciiicsiosloaltaiotinon
sm
1
