All 11 Chapters Covered
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QUIVERS STUVIA
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SOLUTIONS
,Table of Contents
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Acknowledgments ............................................................................................................................. iii
Table of Contents ...............................................................................................................................iv
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CHAPTER 1. INTRODUCTION ....................................................................................................... 1
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1.2 Analytical Versus Numerical Analysis ...................................................................................... 1
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1.4 Applications ............................................................................................................................... 1
Computer Programs ......................................................................................................................... 6
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CHAPTER 2. MATRICES ................................................................................................................. 9
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2.1 Introduction ................................................................................................................................ 9
2.2 Matrix Operations .................................................................................................................... 11
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2.3 Vectors ..................................................................................................................................... 14
2.4 Determinants. ........................................................................................................................... 17
2.5 Rank of a Matrix ...................................................................................................................... 18
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2.6 Applications ............................................................................................................................. 19
CHAPTER 3. INTRODUCTION TO NUMERICAL METHODS. ................................................. 20
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3.1 Introduction .............................................................................................................................. 20
3.2 Accuracy, Precision, and Bias ................................................................................................. 20
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3.3 Significant Figures ................................................................................................................... 22
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3.4 Analysis of Numerical Errors .................................................................................................. 23
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CHAPTER 4. ROOTS OF EQUATIONS ......................................................................................... 27
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4.1 Introduction .............................................................................................................................. 27
4.2 Eigenvalue Analysis ................................................................................................................ 30
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4.3 Direct-Search Method .............................................................................................................. 30
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4.4 Bisection Method. .................................................................................................................... 32
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4.5 Newton-Raphson Iteration. ...................................................................................................... 35
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4.6 Secant Method ......................................................................................................................... 50
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4.8 Synthetic Division.................................................................................................................... 55
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4.9 Multiple Roots ......................................................................................................................... 70
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4.10 Systems of Nonlinear Equations ............................................................................................ 70
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CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS. ............................................................ 72
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5.2 Gaussian Elimination. .............................................................................................................. 72
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5.3 Gauss-Jordan Elimination ........................................................................................................ 74
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5.5 LU Decomposition ................................................................................................................... 76
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5.6 Iterative Equation-Solving Methods. ....................................................................................... 81
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5.6.1 Jacobi Iteration ............................................................................................................................................... 81
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5.6.2 Gaussian-Seidel Iteration ................................................................................................................................ 85
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5.6.3 Convergence Consideration of the Iterative Methods .................................................................................... 90
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5.7 Use of Determinants ................................................................................................................ 94
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5.8 Matrix Inversion ...................................................................................................................... 99
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5.9 Applications ........................................................................................................................... 101
Computer Programs ..................................................................................................................... 103
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CHAPTER 6. NUMERICAL INTERPOLATION ......................................................................... 105
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6.2 Method of Undetermined Coefficients................................................................................... 105
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6.3 Gregory-Newton Interpolation Method ................................................................................. 109
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6.4 Finite Difference Interpolation............................................................................................... 112
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6.5 Newton’s Method .................................................................................................................. 114
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6.6 Lagrange Polynomials ........................................................................................................... 119
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6.7 Interpolation Using Splines ................................................................................................... 124
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6.9 Multi-Dimensional Interpolation ........................................................................................... 133
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CHAPTER 7. DIFFERENTIATION AND IN
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, 7.1 Numerical Differentiation ...................................................................................................... 135
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7.2. Numerical Integration ........................................................................................................... 142
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CHAPTER 8. Differential Equations .............................................................................................. 150
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8.1 Introduction ............................................................................................................................ 150
8.2 Taylor Series Expansion ........................................................................................................ 150
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8.3 Euler’s Method ...................................................................................................................... 154
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8.4 Modified Euler’s Method ....................................................................................................... 157
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8.5 Runge-Kuta Methods ............................................................................................................. 159
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8.6 Predictor-Corrector Methods ................................................................................................. 164
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8.7 Least-Squares Method ........................................................................................................... 167
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8.8 Garlekin Method .................................................................................................................... 170
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8.9 Higher-Order Differential Equations ..................................................................................... 172
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8.10 Boundary Value Problems ................................................................................................... 172
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8.11 Integral Equations ................................................................................................................ 176
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CHAPTER 9. Data Description and Treatment .............................................................................. 177
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9.2 Classification of Data............................................................................................................. 177
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9.3 Graphical Description of Data ............................................................................................... 177
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9.5 Histograms and Frequency Diagrams .................................................................................... 185
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9.6 Descriptive Measures ............................................................................................................. 187
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CHAPTER 10. Curve Fitting and Regression Analysis .................................................................. 190
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10.1 Introduction .......................................................................................................................... 190
10.2 Correlation Analysis ............................................................................................................ 190
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10.3 Introduction to Regression ................................................................................................... 200
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10.4 Principle of Least Squares ................................................................................................... 201
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10.5 Reliability of the Regression Equation ................................................................................ 204
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10.8 Correlation Versus Regression............................................................................................. 207
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10.9 Application of Bivariate Regression Analysis ..................................................................... 209
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10.8 Multiple Regression Analysis .............................................................................................. 213
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10.9 Regression Analysis of Nonlinear Models........................................................................... 220
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CHAPTER 11. Numerical Optimization ......................................................................................... 238
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11.1 Introduction .......................................................................................................................... 238
11.2 The Response Surface Analysis ........................................................................................... 238
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11.3 Numerical Least Squares ..................................................................................................... 239
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11.4 Steepest Descent Method ..................................................................................................... 247
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, CHAPTER 1. INTRODUCTION W W
1.2 Analytical Versus Numerical Analysis W W W
Problem 1-1. W
Solution not provided. W W
Problem 1-2. W
The two methods differ on the basis of their respective algorithms. The analytical method is based
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on analytical calculus while the numerical method is based on finite differences arithmetic.
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Analytical approaches provide direct solutions and will result in exact solutions if they exist. Analy
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tical methods usually require less time to find a solution. Analytical solution procedure becomes c
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onsiderably more complex when constraints are involved. Numerical analysis, on the other hand, c
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an be used to find solutions of moderately complex problems, and it is quite easy to include constra
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ints on the unknowns in the solutions. However, numerical methods most often require a considera
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ble number of iterations in order to find a solution with a reasonable accuracy. The solution provide
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d by the numerical methods is usually not exact. Therefore, error analysis and error estimations are
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required.
1.4 Applications
Problem 1-3. W
2 4
cos( x) 1 x x .......
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2! 4!
For h = 0.1 W W W
x = x0 + h = 0 + 0.1 = 0.1
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cos(0.1) 1.00000000 W W (one term) W
(0.1) 2 W
cos(0.1) 1 (two terms)
0.99500000
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W
2 W
(0.1) 2 (0.1) 4
cos(0.1) 1 0.99500417
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W (three terms)
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2 24
True value = 0.99500417 W W W
The following table summarizes the results for h = 0.1 to 1.0 in an increment of 0.1:
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W W W
QUIVERS STUVIA
W
SOLUTIONS
,Table of Contents
W W
Acknowledgments ............................................................................................................................. iii
Table of Contents ...............................................................................................................................iv
W W
CHAPTER 1. INTRODUCTION ....................................................................................................... 1
W W
1.2 Analytical Versus Numerical Analysis ...................................................................................... 1
W W W
1.4 Applications ............................................................................................................................... 1
Computer Programs ......................................................................................................................... 6
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CHAPTER 2. MATRICES ................................................................................................................. 9
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2.1 Introduction ................................................................................................................................ 9
2.2 Matrix Operations .................................................................................................................... 11
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2.3 Vectors ..................................................................................................................................... 14
2.4 Determinants. ........................................................................................................................... 17
2.5 Rank of a Matrix ...................................................................................................................... 18
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2.6 Applications ............................................................................................................................. 19
CHAPTER 3. INTRODUCTION TO NUMERICAL METHODS. ................................................. 20
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3.1 Introduction .............................................................................................................................. 20
3.2 Accuracy, Precision, and Bias ................................................................................................. 20
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3.3 Significant Figures ................................................................................................................... 22
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3.4 Analysis of Numerical Errors .................................................................................................. 23
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CHAPTER 4. ROOTS OF EQUATIONS ......................................................................................... 27
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4.1 Introduction .............................................................................................................................. 27
4.2 Eigenvalue Analysis ................................................................................................................ 30
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4.3 Direct-Search Method .............................................................................................................. 30
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4.4 Bisection Method. .................................................................................................................... 32
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4.5 Newton-Raphson Iteration. ...................................................................................................... 35
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4.6 Secant Method ......................................................................................................................... 50
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4.8 Synthetic Division.................................................................................................................... 55
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4.9 Multiple Roots ......................................................................................................................... 70
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4.10 Systems of Nonlinear Equations ............................................................................................ 70
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CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS. ............................................................ 72
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5.2 Gaussian Elimination. .............................................................................................................. 72
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5.3 Gauss-Jordan Elimination ........................................................................................................ 74
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5.5 LU Decomposition ................................................................................................................... 76
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5.6 Iterative Equation-Solving Methods. ....................................................................................... 81
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5.6.1 Jacobi Iteration ............................................................................................................................................... 81
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5.6.2 Gaussian-Seidel Iteration ................................................................................................................................ 85
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5.6.3 Convergence Consideration of the Iterative Methods .................................................................................... 90
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5.7 Use of Determinants ................................................................................................................ 94
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5.8 Matrix Inversion ...................................................................................................................... 99
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5.9 Applications ........................................................................................................................... 101
Computer Programs ..................................................................................................................... 103
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CHAPTER 6. NUMERICAL INTERPOLATION ......................................................................... 105
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6.2 Method of Undetermined Coefficients................................................................................... 105
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6.3 Gregory-Newton Interpolation Method ................................................................................. 109
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6.4 Finite Difference Interpolation............................................................................................... 112
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6.5 Newton’s Method .................................................................................................................. 114
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6.6 Lagrange Polynomials ........................................................................................................... 119
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6.7 Interpolation Using Splines ................................................................................................... 124
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6.9 Multi-Dimensional Interpolation ........................................................................................... 133
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CHAPTER 7. DIFFERENTIATION AND IN
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simiciR
sm sioA
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aiotinI
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iv
, 7.1 Numerical Differentiation ...................................................................................................... 135
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7.2. Numerical Integration ........................................................................................................... 142
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CHAPTER 8. Differential Equations .............................................................................................. 150
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8.1 Introduction ............................................................................................................................ 150
8.2 Taylor Series Expansion ........................................................................................................ 150
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8.3 Euler’s Method ...................................................................................................................... 154
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8.4 Modified Euler’s Method ....................................................................................................... 157
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8.5 Runge-Kuta Methods ............................................................................................................. 159
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8.6 Predictor-Corrector Methods ................................................................................................. 164
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8.7 Least-Squares Method ........................................................................................................... 167
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8.8 Garlekin Method .................................................................................................................... 170
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8.9 Higher-Order Differential Equations ..................................................................................... 172
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8.10 Boundary Value Problems ................................................................................................... 172
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8.11 Integral Equations ................................................................................................................ 176
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CHAPTER 9. Data Description and Treatment .............................................................................. 177
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9.2 Classification of Data............................................................................................................. 177
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9.3 Graphical Description of Data ............................................................................................... 177
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9.5 Histograms and Frequency Diagrams .................................................................................... 185
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9.6 Descriptive Measures ............................................................................................................. 187
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CHAPTER 10. Curve Fitting and Regression Analysis .................................................................. 190
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10.1 Introduction .......................................................................................................................... 190
10.2 Correlation Analysis ............................................................................................................ 190
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10.3 Introduction to Regression ................................................................................................... 200
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10.4 Principle of Least Squares ................................................................................................... 201
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10.5 Reliability of the Regression Equation ................................................................................ 204
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10.8 Correlation Versus Regression............................................................................................. 207
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10.9 Application of Bivariate Regression Analysis ..................................................................... 209
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10.8 Multiple Regression Analysis .............................................................................................. 213
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10.9 Regression Analysis of Nonlinear Models........................................................................... 220
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CHAPTER 11. Numerical Optimization ......................................................................................... 238
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11.1 Introduction .......................................................................................................................... 238
11.2 The Response Surface Analysis ........................................................................................... 238
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11.3 Numerical Least Squares ..................................................................................................... 239
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11.4 Steepest Descent Method ..................................................................................................... 247
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SeSiesim
smiciii csiosloaltaiotinon
v
, CHAPTER 1. INTRODUCTION W W
1.2 Analytical Versus Numerical Analysis W W W
Problem 1-1. W
Solution not provided. W W
Problem 1-2. W
The two methods differ on the basis of their respective algorithms. The analytical method is based
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on analytical calculus while the numerical method is based on finite differences arithmetic.
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Analytical approaches provide direct solutions and will result in exact solutions if they exist. Analy
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tical methods usually require less time to find a solution. Analytical solution procedure becomes c
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onsiderably more complex when constraints are involved. Numerical analysis, on the other hand, c
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an be used to find solutions of moderately complex problems, and it is quite easy to include constra
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ints on the unknowns in the solutions. However, numerical methods most often require a considera
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ble number of iterations in order to find a solution with a reasonable accuracy. The solution provide
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d by the numerical methods is usually not exact. Therefore, error analysis and error estimations are
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required.
1.4 Applications
Problem 1-3. W
2 4
cos( x) 1 x x .......
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2! 4!
For h = 0.1 W W W
x = x0 + h = 0 + 0.1 = 0.1
W W W W W W W W W W
cos(0.1) 1.00000000 W W (one term) W
(0.1) 2 W
cos(0.1) 1 (two terms)
0.99500000
W W W W
W
2 W
(0.1) 2 (0.1) 4
cos(0.1) 1 0.99500417
W W
W (three terms)
W W W
2 24
True value = 0.99500417 W W W
The following table summarizes the results for h = 0.1 to 1.0 in an increment of 0.1:
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