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
SeSE G
iesimiciR
sm sioA
iic 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
@@
SeSiesim
smiciii csiosloaltaiotinon
v
, CHAPTER 1. INTRODUCTION
1.2 Analytical Versus Numerical Analysis
Problem 1-1.
Solution not provided.
Problem 1-2.
The ftwo fmethods fdiffer fon fthe fbasis fof ftheir frespective falgorithms. f The fanalytical fmethod fis
fbased fon fanalytical fcalculus fwhile fthe fnumerical fmethod fis fbased fon ffinite fdifferences
farithmetic.
Analytical fapproaches fprovide fdirect fsolutions fand fwill fresult fin fexact fsolutions fif fthey fexist.
fAnalytical fmethods fusually frequire fless ftime fto ffind fa fsolution. f Analytical fsolution fprocedure
fbecomes fconsiderably fmore fcomplex fwhen fconstraints fare finvolved. f Numerical fanalysis, fon
fthe fother fhand, fcan fbe fused fto ffind fsolutions fof fmoderately fcomplex fproblems, fand fit fis fquite
feasy fto finclude fconstraints fon fthe funknowns fin fthe fsolutions. f However, fnumerical fmethods
fmost foften frequire fa fconsiderable fnumber fof fiterations fin forder fto ffind fa fsolution fwith fa
freasonable faccuracy. fThe fsolution fprovided fby fthe fnumerical fmethods fis fusually fnot fexact.
f Therefore, ferror fanalysis fand ferror festimations fare frequired.
1.4 Applications
Problem f1-3.
2 4
cos(fx) f = f1 f−xf + xf −.......
2! 4!
For fh f= f0.1
x f= fx0 f + fh f= f0 f+ f0.1 f= f0.1
cos(0.1) f f1.00000000 (one fterm)
(0.1)f2
cos(0.1) f f1 f− = f0.99500000 (two fterms)
f2
(0.1)f2 (0.1)f4
cos(0.1) f 1 f− + = f0.99500417 (three fterms)
2 24
True fvalue f= f0.99500417
The ffollowing ftable fsummarizes fthe fresults ffor fh f= f0.1 fto f1.0 fin fan fincrement fof f0.1:
@@
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 sioA
iic 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
@@
SeSiesim
smiciii csiosloaltaiotinon
v
, CHAPTER 1. INTRODUCTION
1.2 Analytical Versus Numerical Analysis
Problem 1-1.
Solution not provided.
Problem 1-2.
The ftwo fmethods fdiffer fon fthe fbasis fof ftheir frespective falgorithms. f The fanalytical fmethod fis
fbased fon fanalytical fcalculus fwhile fthe fnumerical fmethod fis fbased fon ffinite fdifferences
farithmetic.
Analytical fapproaches fprovide fdirect fsolutions fand fwill fresult fin fexact fsolutions fif fthey fexist.
fAnalytical fmethods fusually frequire fless ftime fto ffind fa fsolution. f Analytical fsolution fprocedure
fbecomes fconsiderably fmore fcomplex fwhen fconstraints fare finvolved. f Numerical fanalysis, fon
fthe fother fhand, fcan fbe fused fto ffind fsolutions fof fmoderately fcomplex fproblems, fand fit fis fquite
feasy fto finclude fconstraints fon fthe funknowns fin fthe fsolutions. f However, fnumerical fmethods
fmost foften frequire fa fconsiderable fnumber fof fiterations fin forder fto ffind fa fsolution fwith fa
freasonable faccuracy. fThe fsolution fprovided fby fthe fnumerical fmethods fis fusually fnot fexact.
f Therefore, ferror fanalysis fand ferror festimations fare frequired.
1.4 Applications
Problem f1-3.
2 4
cos(fx) f = f1 f−xf + xf −.......
2! 4!
For fh f= f0.1
x f= fx0 f + fh f= f0 f+ f0.1 f= f0.1
cos(0.1) f f1.00000000 (one fterm)
(0.1)f2
cos(0.1) f f1 f− = f0.99500000 (two fterms)
f2
(0.1)f2 (0.1)f4
cos(0.1) f 1 f− + = f0.99500417 (three fterms)
2 24
True fvalue f= f0.99500417
The ffollowing ftable fsummarizes fthe fresults ffor fh f= f0.1 fto f1.0 fin fan fincrement fof f0.1:
@@
SeSiesimiciiicsiosloaltaiotinon
sm
1
