Solution Manual for Numerical Analysis for Engineers: Methods and Applications (2nd Edition by Ayyub) – Complete Problem Solutions

All11ChaptersCovered
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SOLUTIONS

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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 l




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-SeidelIteration ................................................................................................................................. 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-NewtonInterpolationMethod ................................................................................... 109 l l




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 l @@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 l




8.7 Least-SquaresMethod ............................................................................................................ 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 l l




1.2 Analytical Versus Numerical Analysis l l l




Problem 1-1. l




Solution not provided. l l




Problem 1-2. l




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

Problem 1-3. l



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




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




(0.1)2 l




cos(0.1)  1− =0.99500000 (two terms) l l l l l



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(0.1) (0.1)4
cos(0.1) 1− + = 0.99500417
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(three terms) l l l l




2 24
True value = 0.99500417 l l l




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