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