Numerical Methods with Worked ExamplesThis book is for students following a module in numerical methods, numerical techniques, or numerical analysis. It approaches the subject from a pragmatic viewpoint, appropriate for the modern student. The theory is kept to a minimum commensurate with comprehensive coverage of the subject and it contains abundant worked examples which provide easy understanding through a clear and concise theoretical treatment. |
Contents
Linear equations | 1 |
11 INTRODUCTION | 2 |
13 GAUSSIAN ELIMINATION | 6 |
14 SINGULAR SYSTEMS | 16 |
15 SYMMETRIC POSITIVE DEFINITE SYSTEMS | 18 |
16 ITERATIVE REFINEMENT | 19 |
17 ITERATIVE METHODS FOR SPARSE SYSTEMS | 24 |
18 EXERCISES | 29 |
65 INTEGER PROGRAMMING | 144 |
66 DECISION PROBLEMS | 148 |
67 THE TRAVELLING SALESMAN PROBLEM | 150 |
68 THE MACHINE SCHEDULING PROBLEM | 152 |
69 EXERCISES | 156 |
Optimization | 163 |
71 INTRODUCTION | 164 |
73 GOLDEN SECTION SEARCH | 167 |
Nonlinear equations | 35 |
21 INTRODUCTION | 36 |
22 BISECTION METHOD | 37 |
23 RULE OF FALSE POSITION | 40 |
24 THE SECANT METHOD | 42 |
25 THE BUS AND DEKKER METHOD | 45 |
26 NEWTONRAPHSON METHOD | 47 |
27 COMPARISON OF METHODS FOR A SINGLE EQUATION | 50 |
28 NEWTONS METHOD FOR SYSTEMS OF NONLINEAR EQUATIONS | 51 |
29 EXERCISES | 58 |
Curve fitting | 63 |
32 LINEAR INTERPOLATION | 64 |
33 POLYNOMIAL INTERPOLATION | 70 |
34 LEAST SQUARES APPROXIMATION | 79 |
35 EXERCISES | 85 |
Numerical integration | 91 |
41 INTRODUCTION | 92 |
43 INTEGRATION OF FUNCTIONS | 98 |
44 HIGHER ORDER RULES | 103 |
45 ADAPTIVE QUADRATURE | 105 |
46 EXERCISES | 107 |
Numerical differentiation | 111 |
51 INTRODUCTION | 112 |
53 THREEAND FIVEPOINT FORMULAE | 114 |
54 HIGHER ORDER DERIVATIVES | 117 |
55 CAUCHYS THEOREM | 120 |
56 EXERCISES | 123 |
Linear programming | 129 |
61 INTRODUCTION | 130 |
63 CANONICAL FORM | 134 |
64 THE SIMPLEX METHOD | 136 |
74 MINIMIZATION STRATEGY FOR UNCONSTRAINED PROBLEMS | 170 |
76 A RANKONE METHOD | 174 |
77 CONSTRAINED OPTIMIZATION | 181 |
78 MINIMIZATION BY USE OF A SIMPLE PENALTY FUNCTION | 182 |
79 MINIMIZATION USING A LAGRANGIAN | 184 |
710 A PENALTY FUNCTION FOR INEQUALITY CONSTRAINTS | 187 |
711 EXERCISES | 190 |
Ordinary differential equations | 195 |
81 INTRODUCTION | 196 |
82 FIRST ORDER EQUATIONS | 198 |
83 HIGHER ORDER EQUATIONS | 209 |
84 BOUNDARY VALUE PROBLEMS | 212 |
85 FINITE DIFFERENCES | 215 |
86 ACCURACY | 217 |
87 EXERCISES | 219 |
Eigenvalues and eigenvectors | 227 |
92 THE CHARACTERISTIC POLYNOMIAL | 229 |
93 THE POWER METHOD | 231 |
94 EIGENVALUES OF SPECIAL MATRICES | 235 |
95 A SIMPLE QR METHOD | 237 |
96 EXERCISES | 242 |
Statistics | 247 |
101 INTRODUCTION | 248 |
103 LEAST SQUARES ANALYSIS | 256 |
104 RANDOM NUMBERS | 260 |
105 RANDOM NUMBER GENERATORS | 261 |
106 MONTECARLO QUADRATURE | 264 |
107 EXERCISES | 265 |
| 269 | |
| 271 | |
Common terms and phrases
accuracy analytic approximation arithmetic backward substitution bisection method branch and bound Bus and Dekker calculations chapter column consider constraints convergence corresponding eigenvector data points decimal places differences differential equation Discussion eigenvalues eigenvector error estimate Euler's method example feasible region feasible vertex five-point formula function evaluations garment factory problem gas problem Gaussian elimination given gives golden section search initial-value problem integral integrand interpolation involved iterative refinement least squares linear equations linear programming problem linear system local minimum method of steepest method to find minimum Newton-Raphson method Newton's method nonlinear obtain partial pivoting penalty function method polynomial Problem Solution produce quadratic quadrature R₁ random number rank-one method reduce Repeat from step right-hand side Runge-Kutta method secant method sequence shown simplex method Simpson's rule solve steepest descent straight line system of equations trapezium rule upper triangular form zero