Solution Manual for Numerical Analysis for Engineers: Methods and Applications (2nd Edition, 2016) by Ayyub – Covers All 11 Chapters

All11ChaptersCovered
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SOLUTIONS

,Table of Contents b b




Acknowledgments ............................................................................................................................. iii
Table of Contents ................................................................................................................................iv
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CHAPTER 1. INTRODUCTION ........................................................................................................ 1
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1.2 Analytical Versus Numerical Analysis ....................................................................................... 1
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1.4 Applications ............................................................................................................................... 1
Computer Programs.......................................................................................................................... 6
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CHAPTER 2. MATRICES .................................................................................................................. 9
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2.1 Introduction ................................................................................................................................ 9
2.2 Matrix Operations..................................................................................................................... 11
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2.3 Vectors ..................................................................................................................................... 14
2.4 Determinants. ........................................................................................................................... 17
2.5 Rank of a Matrix ....................................................................................................................... 18
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2.6 Applications ............................................................................................................................. 19
CHAPTER 3. INTRODUCTION TO NUMERICAL METHODS. .................................................... 20
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3.1 Introduction .............................................................................................................................. 20
3.2 Accuracy, Precision, and Bias ................................................................................................... 20
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3.3 Significant Figures ................................................................................................................... 22
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3.4 Analysis of Numerical Errors .................................................................................................... 23
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CHAPTER 4. ROOTS OF EQUATIONS ........................................................................................... 27
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4.1 Introduction .............................................................................................................................. 27
4.2 Eigenvalue Analysis ................................................................................................................. 30
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4.3 Direct-Search Method .............................................................................................................. 30
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4.4 Bisection Method. .................................................................................................................... 32
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4.5 Newton-Raphson Iteration. ...................................................................................................... 35
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4.6 Secant Method .......................................................................................................................... 50
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4.8 Synthetic Division .................................................................................................................... 55
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4.9 Multiple Roots.......................................................................................................................... 70
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4.10 Systems of Nonlinear Equations ............................................................................................. 70
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CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS. .............................................................. 72
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5.2 Gaussian Elimination. .............................................................................................................. 72
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5.3 Gauss-Jordan Elimination ........................................................................................................ 74
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5.5 LU Decomposition ................................................................................................................... 76
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5.6 Iterative Equation-Solving Methods. ........................................................................................ 81
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5.6.1 Jacobi Iteration ................................................................................................................................................ 81
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5.6.2 Gaussian-Seidel Iteration................................................................................................................................. 85
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5.6.3 Convergence Consideration of the Iterative Methods ....................................................................................... 90
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5.7 Use of Determinants ................................................................................................................. 94
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5.8 Matrix Inversion ....................................................................................................................... 99
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5.9 Applications ........................................................................................................................... 101
Computer Programs...................................................................................................................... 103
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CHAPTER 6. NUMERICAL INTERPOLATION ........................................................................... 105
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6.2 Method of Undetermined Coefficients .................................................................................... 105
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6.3 Gregory-Newton Interpolation Method................................................................................... 109
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6.4 Finite Difference Interpolation ............................................................................................... 112
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6.5 Newton’s Method ................................................................................................................... 114
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6.6 Lagrange Polynomials ............................................................................................................ 119
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6.7 Interpolation Using Splines .................................................................................................... 124
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6.9 Multi-Dimensional Interpolation ............................................................................................ 133
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CHAPTER 7. DIFFERENTIATION AND IN b b @@T
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, 7.1 Numerical Differentiation ...................................................................................................... 135
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7.2. Numerical Integration............................................................................................................ 142
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CHAPTER 8. Differential Equations ............................................................................................... 150
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8.1 Introduction ............................................................................................................................ 150
8.2 Taylor Series Expansion ......................................................................................................... 150
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8.3 Euler’s Method ....................................................................................................................... 154
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8.4 Modified Euler’s Method ........................................................................................................ 157
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8.5 Runge-Kuta Methods ............................................................................................................. 159
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8.6 Predictor-Corrector Methods.................................................................................................. 164
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8.7 Least-Squares Method ............................................................................................................ 167
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8.8 Garlekin Method .................................................................................................................... 170
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8.9 Higher-Order Differential Equations ...................................................................................... 172
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8.10 Boundary Value Problems .................................................................................................... 172
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8.11 Integral Equations ................................................................................................................ 176
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CHAPTER 9. Data Description and Treatment ................................................................................. 177
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9.2 Classification of Data ............................................................................................................. 177
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9.3 Graphical Description of Data................................................................................................. 177
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9.5 Histograms and Frequency Diagrams ..................................................................................... 185
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9.6 Descriptive Measures ............................................................................................................. 187
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CHAPTER 10. Curve Fitting and Regression Analysis .................................................................... 190
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10.1 Introduction .......................................................................................................................... 190
10.2 Correlation Analysis ............................................................................................................. 190
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10.3 Introduction to Regression .................................................................................................... 200
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10.4 Principle of Least Squares ..................................................................................................... 201
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10.5 Reliability of the Regression Equation .................................................................................. 204
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10.8 Correlation Versus Regression.............................................................................................. 207
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10.9 Application of Bivariate Regression Analysis ....................................................................... 209
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10.8 Multiple Regression Analysis ............................................................................................... 213
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10.9 Regression Analysis of Nonlinear Models ............................................................................ 220
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CHAPTER 11. Numerical Optimization .......................................................................................... 238
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11.1 Introduction .......................................................................................................................... 238
11.2 The Response Surface Analysis ............................................................................................ 238
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11.3 Numerical Least Squares ...................................................................................................... 239
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11.4 Steepest Descent Method ...................................................................................................... 247
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, CHAPTER 1. INTRODUCTION b b




1.2 Analytical Versus Numerical Analysis b b b




Problem 1-1. b




Solution not provided.b b




Problem 1-2. b




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.
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Analytical methods usually require less time to find a solution. Analytical solution procedure
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becomes considerably more complex when constraints are involved. Numerical analysis, on the other
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hand, can be used to find solutions of moderately complex problems, and it is quite easy to include
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constraints on the unknowns in the solutions. However, numerical methods most often require a
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considerable number of iterations in order to find a solution with a reasonable accuracy. The solution
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provided by the numerical methods is usually not exact. Therefore, error analysis and error estimations
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are required.
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1.4 Applications

Problem 1-3. b



2 4
cos(x) = 1− x + x −.......
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2! 4!
For h = 0.1 b b b




x = x0 + h = 0 + 0.1 = 0.1
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cos(0.1)  1.00000000 (one term)
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(0.1)2 b




cos(0.1)  1 − = 0.99500000 (two terms)
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22 4
b




(0.1) (0.1)
cos(0.1) 1− + = 0.99500417
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(three terms)
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2 24
True value = 0.99500417 b b b




The following table summarizes the results for h = 0.1 to 1.0 in an increment of 0.1:
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