All 11 Chapters Covered
oo oo oo
SOLUTIONS
,Table of Contents
oo oo
Acknowledgments ......................................................................................................................... iii
Table ooof ooContents ....................................................................................................................... iv
CHAPTER oo1. ooINTRODUCTION ................................................................................................ 1
1.2 Analytical ooVersus ooNumerical ooAnalysis .............................................................................. 1
1.4 Applications .......................................................................................................................... 1
Computer ooPrograms ................................................................................................................... 6
CHAPTER oo2. ooMATRICES .......................................................................................................... 9
2.1 Introduction........................................................................................................................... 9
2.2 Matrix ooOperations .............................................................................................................. 11
2.3 Vectors ............................................................................................................................... 14
2.4 Determinants. ...................................................................................................................... 17
2.5 Rank ooof ooa ooMatrix............................................................................................................. 18
2.6 Applications ........................................................................................................................ 19
CHAPTER oo3. ooINTRODUCTION ooTO ooNUMERICAL ooMETHODS. ......................................... 20
3.1 Introduction......................................................................................................................... 20
3.2 Accuracy, ooPrecision, ooand ooBias ......................................................................................... 20
3.3 Significant ooFigures ............................................................................................................. 22
3.4 Analysis ooof ooNumerical ooErrors .......................................................................................... 23
CHAPTER oo4. ooROOTS ooOF ooEQUATIONS ................................................................................ 27
4.1 Introduction......................................................................................................................... 27
4.2 Eigenvalue ooAnalysis .......................................................................................................... 30
4.3 Direct-Search ooMethod........................................................................................................ 30
4.4 Bisection ooMethod............................................................................................................... 32
4.5 Newton-Raphson ooIteration. ................................................................................................ 35
4.6 Secant ooMethod ................................................................................................................... 50
4.8 Synthetic ooDivision ............................................................................................................. 55
4.9 Multiple ooRoots ................................................................................................................... 70
4.10 Systems ooof ooNonlinear ooEquations .................................................................................... 70
CHAPTER oo5. ooSIMULTANEOUS ooLINEAR ooEQUATIONS. .................................................... 72
5.2 Gaussian ooElimination. ........................................................................................................ 72
5.3 Gauss-Jordan ooElimination .................................................................................................. 74
5.5 LU ooDecomposition ............................................................................................................. 76
5.6 Iterative ooEquation-Solving ooMethods.................................................................................. 81
5.6.1 Jacobi ooIteration....................................................................................................................................... 81
5.6.2 Gaussian-Seidel ooIteration ........................................................................................................................ 85
5.6.3 Convergence ooConsideration ooof oothe ooIterative ooMethods ........................................................................ 90
5.7 Use ooof ooDeterminants ......................................................................................................... 94
5.8 Matrix ooInversion ................................................................................................................ 99
5.9 Applications ...................................................................................................................... 101
Computer ooPrograms ............................................................................................................... 103
CHAPTER oo6. ooNUMERICAL ooINTERPOLATION .................................................................. 105
6.2 Method ooof ooUndetermined ooCoefficients ........................................................................... 105
6.3 Gregory-Newton ooInterpolation ooMethod ............................................................................ 109
6.4 Finite ooDifference ooInterpolation........................................................................................ 112
6.5 Newton’s ooMethod............................................................................................................. 114
6.6 Lagrange ooPolynomials ..................................................................................................... 119
6.7 Interpolation ooUsing ooSplines ............................................................................................ 124
6.9 ooMulti-Dimensional ooInterpolation..................................................................................... 133
CHAPTER oo7. ooDIFFERENTIATION ooAND ooIN @@T
SeSE G
iesimiciR
sm iicsioA
sloaltT
aiotinI
oOn o o N.............................................. 135
iv
, 7.1 Numerical ooDifferentiation ................................................................................................ 135
7.2. Numerical ooIntegration ..................................................................................................... 142
CHAPTER oo8. ooDifferential ooEquations...................................................................................... 150
8.1 Introduction....................................................................................................................... 150
8.2 Taylor ooSeries ooExpansion ................................................................................................. 150
8.3 Euler’s ooMethod ................................................................................................................ 154
8.4 Modified ooEuler’s ooMethod ................................................................................................ 157
8.5 Runge-Kuta ooMethods ....................................................................................................... 159
8.6 Predictor-Corrector ooMethods ............................................................................................ 164
8.7 Least-Squares ooMethod ...................................................................................................... 167
8.8 Garlekin ooMethod.............................................................................................................. 170
8.9 Higher-Order ooDifferential ooEquations .............................................................................. 172
8.10 Boundary ooValue ooProblems ............................................................................................ 172
8.11 Integral ooEquations .......................................................................................................... 176
CHAPTER oo9. ooData ooDescription ooand ooTreatment .................................................................... 177
9.2 Classification ooof ooData ..................................................................................................... 177
9.3 Graphical ooDescription ooof ooData....................................................................................... 177
9.5 Histograms ooand ooFrequency ooDiagrams ............................................................................ 185
9.6 Descriptive ooMeasures....................................................................................................... 187
CHAPTER oo10. ooCurve ooFitting ooand ooRegression ooAnalysis ...................................................... 190
10.1 Introduction..................................................................................................................... 190
10.2 Correlation ooAnalysis ...................................................................................................... 190
10.3 Introduction ooto ooRegression ............................................................................................ 200
10.4 Principle ooof ooLeast ooSquares ........................................................................................... 201
10.5 Reliability ooof oothe ooRegression ooEquation ....................................................................... 204
10.8 Correlation ooVersus ooRegression ...................................................................................... 207
10.9 Application ooof ooBivariate ooRegression ooAnalysis ............................................................ 209
10.8 Multiple ooRegression ooAnalysis ....................................................................................... 213
10.9 Regression ooAnalysis ooof ooNonlinear ooModels .................................................................. 220
CHAPTER oo11. ooNumerical ooOptimization................................................................................. 238
11.1 Introduction..................................................................................................................... 238
11.2 The ooResponse ooSurface ooAnalysis .................................................................................. 238
11.3 Numerical ooLeast ooSquares .............................................................................................. 239
11.4 Steepest ooDescent ooMethod.............................................................................................. 247
@@
SeSies im
smiciii cs iosloaltaiotinon
v
, CHAPTER 1. INTRODUCTION
oo oo
1.2 Analytical Versus Numerical Analysis
oo oo oo
Problem oo1-1.
Solution oonot ooprovided.
Problem oo1-2.
The ootwo oomethods oodiffer ooon oothe oobasis ooof ootheir oorespective ooalgorithms. o o The ooanalytical
oomethod oois oobased ooon ooanalytical oocalculus oowhile oothe oonumerical oomethod oois oobased ooon oofinite
oodifferences ooarithmetic.
Analytical ooapproaches ooprovide oodirect oosolutions ooand oowill ooresult ooin ooexact oosolutions ooif oothey
ooexist. ooAnalytical oomethods oousually oorequire ooless ootime ooto oofind ooa oosolution. o o Analytical
oosolution ooprocedure oobecomes ooconsiderably oomore oocomplex oowhen ooconstraints ooare ooinvolved.
o o Numerical ooanalysis, ooon oothe ooother oohand, oocan oobe ooused ooto oofind oosolutions ooof oomoderately
oocomplex ooproblems, ooand ooit oois ooquite ooeasy ooto ooinclude ooconstraints ooon oothe oounknowns ooin oothe
oosolutions. o o However, oonumerical oomethods oomost oooften oorequire ooa ooconsiderable oonumber ooof
ooiterations ooin ooorder ooto oofind ooa oosolution oowith ooa ooreasonable ooaccuracy. ooThe oosolution
ooprovided ooby oothe oonumerical oomethods oois oousually oonot ooexact. o o Therefore, ooerror ooanalysis ooand
ooerror ooestimations ooare oorequired.
1.4 Applications
Problem oo1-3.
2 4
cos(oox) o o = oo1oox−oo + xoo −.......
2! 4!
For ooh oo= oo0.1
x oo= oox0 o o + ooh oo= oo0 oo+ oo0.1 oo= oo0.1
cos(0.1) oo oo1.00000000 (one ooterm)
(0.1)oo2
cos(0.1) oo oo1 oo− = (two ooterms)
oo0.99500000 oo2
2 4
(0.1)oo (0.1)oo
cos(0.1) o o 1oo− + = (three ooterms)
oo 0.99500417
2 24
True oovalue oo= oo0.99500417
The oofollowing ootable oosummarizes oothe ooresults oofor ooh oo= oo0.1 ooto oo1.0 ooin ooan ooincrement ooof oo0.1:
@@
SeSiesimiciiicsiosloaltaiotinon
sm
1
oo oo oo
SOLUTIONS
,Table of Contents
oo oo
Acknowledgments ......................................................................................................................... iii
Table ooof ooContents ....................................................................................................................... iv
CHAPTER oo1. ooINTRODUCTION ................................................................................................ 1
1.2 Analytical ooVersus ooNumerical ooAnalysis .............................................................................. 1
1.4 Applications .......................................................................................................................... 1
Computer ooPrograms ................................................................................................................... 6
CHAPTER oo2. ooMATRICES .......................................................................................................... 9
2.1 Introduction........................................................................................................................... 9
2.2 Matrix ooOperations .............................................................................................................. 11
2.3 Vectors ............................................................................................................................... 14
2.4 Determinants. ...................................................................................................................... 17
2.5 Rank ooof ooa ooMatrix............................................................................................................. 18
2.6 Applications ........................................................................................................................ 19
CHAPTER oo3. ooINTRODUCTION ooTO ooNUMERICAL ooMETHODS. ......................................... 20
3.1 Introduction......................................................................................................................... 20
3.2 Accuracy, ooPrecision, ooand ooBias ......................................................................................... 20
3.3 Significant ooFigures ............................................................................................................. 22
3.4 Analysis ooof ooNumerical ooErrors .......................................................................................... 23
CHAPTER oo4. ooROOTS ooOF ooEQUATIONS ................................................................................ 27
4.1 Introduction......................................................................................................................... 27
4.2 Eigenvalue ooAnalysis .......................................................................................................... 30
4.3 Direct-Search ooMethod........................................................................................................ 30
4.4 Bisection ooMethod............................................................................................................... 32
4.5 Newton-Raphson ooIteration. ................................................................................................ 35
4.6 Secant ooMethod ................................................................................................................... 50
4.8 Synthetic ooDivision ............................................................................................................. 55
4.9 Multiple ooRoots ................................................................................................................... 70
4.10 Systems ooof ooNonlinear ooEquations .................................................................................... 70
CHAPTER oo5. ooSIMULTANEOUS ooLINEAR ooEQUATIONS. .................................................... 72
5.2 Gaussian ooElimination. ........................................................................................................ 72
5.3 Gauss-Jordan ooElimination .................................................................................................. 74
5.5 LU ooDecomposition ............................................................................................................. 76
5.6 Iterative ooEquation-Solving ooMethods.................................................................................. 81
5.6.1 Jacobi ooIteration....................................................................................................................................... 81
5.6.2 Gaussian-Seidel ooIteration ........................................................................................................................ 85
5.6.3 Convergence ooConsideration ooof oothe ooIterative ooMethods ........................................................................ 90
5.7 Use ooof ooDeterminants ......................................................................................................... 94
5.8 Matrix ooInversion ................................................................................................................ 99
5.9 Applications ...................................................................................................................... 101
Computer ooPrograms ............................................................................................................... 103
CHAPTER oo6. ooNUMERICAL ooINTERPOLATION .................................................................. 105
6.2 Method ooof ooUndetermined ooCoefficients ........................................................................... 105
6.3 Gregory-Newton ooInterpolation ooMethod ............................................................................ 109
6.4 Finite ooDifference ooInterpolation........................................................................................ 112
6.5 Newton’s ooMethod............................................................................................................. 114
6.6 Lagrange ooPolynomials ..................................................................................................... 119
6.7 Interpolation ooUsing ooSplines ............................................................................................ 124
6.9 ooMulti-Dimensional ooInterpolation..................................................................................... 133
CHAPTER oo7. ooDIFFERENTIATION ooAND ooIN @@T
SeSE G
iesimiciR
sm iicsioA
sloaltT
aiotinI
oOn o o N.............................................. 135
iv
, 7.1 Numerical ooDifferentiation ................................................................................................ 135
7.2. Numerical ooIntegration ..................................................................................................... 142
CHAPTER oo8. ooDifferential ooEquations...................................................................................... 150
8.1 Introduction....................................................................................................................... 150
8.2 Taylor ooSeries ooExpansion ................................................................................................. 150
8.3 Euler’s ooMethod ................................................................................................................ 154
8.4 Modified ooEuler’s ooMethod ................................................................................................ 157
8.5 Runge-Kuta ooMethods ....................................................................................................... 159
8.6 Predictor-Corrector ooMethods ............................................................................................ 164
8.7 Least-Squares ooMethod ...................................................................................................... 167
8.8 Garlekin ooMethod.............................................................................................................. 170
8.9 Higher-Order ooDifferential ooEquations .............................................................................. 172
8.10 Boundary ooValue ooProblems ............................................................................................ 172
8.11 Integral ooEquations .......................................................................................................... 176
CHAPTER oo9. ooData ooDescription ooand ooTreatment .................................................................... 177
9.2 Classification ooof ooData ..................................................................................................... 177
9.3 Graphical ooDescription ooof ooData....................................................................................... 177
9.5 Histograms ooand ooFrequency ooDiagrams ............................................................................ 185
9.6 Descriptive ooMeasures....................................................................................................... 187
CHAPTER oo10. ooCurve ooFitting ooand ooRegression ooAnalysis ...................................................... 190
10.1 Introduction..................................................................................................................... 190
10.2 Correlation ooAnalysis ...................................................................................................... 190
10.3 Introduction ooto ooRegression ............................................................................................ 200
10.4 Principle ooof ooLeast ooSquares ........................................................................................... 201
10.5 Reliability ooof oothe ooRegression ooEquation ....................................................................... 204
10.8 Correlation ooVersus ooRegression ...................................................................................... 207
10.9 Application ooof ooBivariate ooRegression ooAnalysis ............................................................ 209
10.8 Multiple ooRegression ooAnalysis ....................................................................................... 213
10.9 Regression ooAnalysis ooof ooNonlinear ooModels .................................................................. 220
CHAPTER oo11. ooNumerical ooOptimization................................................................................. 238
11.1 Introduction..................................................................................................................... 238
11.2 The ooResponse ooSurface ooAnalysis .................................................................................. 238
11.3 Numerical ooLeast ooSquares .............................................................................................. 239
11.4 Steepest ooDescent ooMethod.............................................................................................. 247
@@
SeSies im
smiciii cs iosloaltaiotinon
v
, CHAPTER 1. INTRODUCTION
oo oo
1.2 Analytical Versus Numerical Analysis
oo oo oo
Problem oo1-1.
Solution oonot ooprovided.
Problem oo1-2.
The ootwo oomethods oodiffer ooon oothe oobasis ooof ootheir oorespective ooalgorithms. o o The ooanalytical
oomethod oois oobased ooon ooanalytical oocalculus oowhile oothe oonumerical oomethod oois oobased ooon oofinite
oodifferences ooarithmetic.
Analytical ooapproaches ooprovide oodirect oosolutions ooand oowill ooresult ooin ooexact oosolutions ooif oothey
ooexist. ooAnalytical oomethods oousually oorequire ooless ootime ooto oofind ooa oosolution. o o Analytical
oosolution ooprocedure oobecomes ooconsiderably oomore oocomplex oowhen ooconstraints ooare ooinvolved.
o o Numerical ooanalysis, ooon oothe ooother oohand, oocan oobe ooused ooto oofind oosolutions ooof oomoderately
oocomplex ooproblems, ooand ooit oois ooquite ooeasy ooto ooinclude ooconstraints ooon oothe oounknowns ooin oothe
oosolutions. o o However, oonumerical oomethods oomost oooften oorequire ooa ooconsiderable oonumber ooof
ooiterations ooin ooorder ooto oofind ooa oosolution oowith ooa ooreasonable ooaccuracy. ooThe oosolution
ooprovided ooby oothe oonumerical oomethods oois oousually oonot ooexact. o o Therefore, ooerror ooanalysis ooand
ooerror ooestimations ooare oorequired.
1.4 Applications
Problem oo1-3.
2 4
cos(oox) o o = oo1oox−oo + xoo −.......
2! 4!
For ooh oo= oo0.1
x oo= oox0 o o + ooh oo= oo0 oo+ oo0.1 oo= oo0.1
cos(0.1) oo oo1.00000000 (one ooterm)
(0.1)oo2
cos(0.1) oo oo1 oo− = (two ooterms)
oo0.99500000 oo2
2 4
(0.1)oo (0.1)oo
cos(0.1) o o 1oo− + = (three ooterms)
oo 0.99500417
2 24
True oovalue oo= oo0.99500417
The oofollowing ootable oosummarizes oothe ooresults oofor ooh oo= oo0.1 ooto oo1.0 ooin ooan ooincrement ooof oo0.1:
@@
SeSiesimiciiicsiosloaltaiotinon
sm
1
