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
iv
,CHAPTER 7. DIFFERENTIATION AND IN@@TSEeSiesGimsmiciiRicsAsioloaltTaiotiIonOn N............................................................................................................................................. 135
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
v
, SeSise mismiciiisc iosloaltaiotinon
@@
vi
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
iv
,CHAPTER 7. DIFFERENTIATION AND IN@@TSEeSiesGimsmiciiRicsAsioloaltTaiotiIonOn N............................................................................................................................................. 135
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
v
, SeSise mismiciiisc iosloaltaiotinon
@@
vi
