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