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
iv
, 6.9 Multi-Dimensional Interpolation................................................................................................... 133
CHAPTER 7. DIFFERENTIATION AND IN @@TSeSEiesimGiciR
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7.1 Numerical Differentiation ................................................................................................................ 135
7.2. Numerical Integration ...................................................................................................................... 142
CHAPTER 8. Differential Equations ........................................................................................................ 150
8.1 Introduction ........................................................................................................................................... 150
8.2 Taylor Series Expansion .................................................................................................................. 150
8.3 Euler’s Method ..................................................................................................................................... 154
8.4 Modified Euler’s Method ................................................................................................................... 157
8.5 Runge-Kuta Methods ........................................................................................................................ 159
8.6 Predictor-Corrector Methods ........................................................................................................ 164
8.7 Least-Squares Method ..................................................................................................................... 167
8.8 Garlekin Method .................................................................................................................................. 170
8.9 Higher-Order Differential Equations........................................................................................... 172
8.10 Boundary Value Problems ............................................................................................................ 172
8.11 Integral Equations ............................................................................................................................ 176
CHAPTER 9. Data Description and Treatment .................................................................................... 177
9.2 Classification of Data ........................................................................................................................ 177
9.3 Graphical Description of Data........................................................................................................ 177
9.5 Histograms and Frequency Diagrams........................................................................................ 185
9.6 Descriptive Measures ....................................................................................................................... 187
CHAPTER 10. Curve Fitting and Regression Analysis ..................................................................... 190
10.1 Introduction......................................................................................................................................... 190
10.2 Correlation Analysis ....................................................................................................................... 190
10.3 Introduction to Regression .......................................................................................................... 200
10.4 Principle of Least Squares ........................................................................................................... 201
10.5 Reliability of the Regression Equation .................................................................................... 204
10.8 Correlation Versus Regression ................................................................................................. 207
10.9 Application of Bivariate Regression Analysis ...................................................................... 209
10.8 Multiple Regression Analysis ...................................................................................................... 213
10.9 Regression Analysis of Nonlinear Models ............................................................................ 220
CHAPTER 11. Numerical Optimization ................................................................................................... 238
11.1 Introduction........................................................................................................................................ 238
11.2 The Response Surface Analysis ................................................................................................ 238
11.3 Numerical Least Squares ............................................................................................................ 239
11.4 Steepest Descent Method ............................................................................................................ 247
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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
iv
, 6.9 Multi-Dimensional Interpolation................................................................................................... 133
CHAPTER 7. DIFFERENTIATION AND IN @@TSeSEiesimGiciR
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7.1 Numerical Differentiation ................................................................................................................ 135
7.2. Numerical Integration ...................................................................................................................... 142
CHAPTER 8. Differential Equations ........................................................................................................ 150
8.1 Introduction ........................................................................................................................................... 150
8.2 Taylor Series Expansion .................................................................................................................. 150
8.3 Euler’s Method ..................................................................................................................................... 154
8.4 Modified Euler’s Method ................................................................................................................... 157
8.5 Runge-Kuta Methods ........................................................................................................................ 159
8.6 Predictor-Corrector Methods ........................................................................................................ 164
8.7 Least-Squares Method ..................................................................................................................... 167
8.8 Garlekin Method .................................................................................................................................. 170
8.9 Higher-Order Differential Equations........................................................................................... 172
8.10 Boundary Value Problems ............................................................................................................ 172
8.11 Integral Equations ............................................................................................................................ 176
CHAPTER 9. Data Description and Treatment .................................................................................... 177
9.2 Classification of Data ........................................................................................................................ 177
9.3 Graphical Description of Data........................................................................................................ 177
9.5 Histograms and Frequency Diagrams........................................................................................ 185
9.6 Descriptive Measures ....................................................................................................................... 187
CHAPTER 10. Curve Fitting and Regression Analysis ..................................................................... 190
10.1 Introduction......................................................................................................................................... 190
10.2 Correlation Analysis ....................................................................................................................... 190
10.3 Introduction to Regression .......................................................................................................... 200
10.4 Principle of Least Squares ........................................................................................................... 201
10.5 Reliability of the Regression Equation .................................................................................... 204
10.8 Correlation Versus Regression ................................................................................................. 207
10.9 Application of Bivariate Regression Analysis ...................................................................... 209
10.8 Multiple Regression Analysis ...................................................................................................... 213
10.9 Regression Analysis of Nonlinear Models ............................................................................ 220
CHAPTER 11. Numerical Optimization ................................................................................................... 238
11.1 Introduction........................................................................................................................................ 238
11.2 The Response Surface Analysis ................................................................................................ 238
11.3 Numerical Least Squares ............................................................................................................ 239
11.4 Steepest Descent Method ............................................................................................................ 247
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