Solution Manual to Numerical Analysis for Engineers: Methods and Applications (Second Edition) by Bilal M. Ayyub and Richard H. McCuen:

All 11 Chapters Covered
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SOLUTIONS

,Table of Contents
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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
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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
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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 fg fg




1.2 Analytical Versus Numerical Analysis fg fg fg




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Problem 1-2. fg


The two methods differ on the basis of their respective algorithms. The analytical method is
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based on analytical calculus while the numerical method is based on finite differences
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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
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the other hand, can be used to find solutions of moderately complex problems, and it is quite
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easy to include constraints on the unknowns in the solutions. However, numerical methods
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most often require a considerable number of iterations in order to find a solution with a
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reasonable accuracy. The solution provided by the numerical methods is usually not exact.
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Therefore, error analysis and error estimations are required.
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1.4 Applications

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cos(x)  1 x  x .......
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For h = 0.1 fg fg fg


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


cos(0.1)  1   0.99500000 fg (two terms)
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(0.1) (0.1)4 fg fg

cos(0.1) 1   0.99500417 fg (three terms)
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2 24
True value = 0.99500417 fg fg fg


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