Analysis for Computer Scientists. Foundations, Methods and Algorithms [2nd ed.]

Description

Contents......Page 6 1.1 The Real Numbers......Page 10 1.2 Order Relation and Arithmetic on mathbbR......Page 14 1.3 Machine Numbers......Page 17 1.4 Rounding......Page 19 1.5 Exercises......Page 20 2.1 Basic Notions......Page 22 2.2 Some Elementary Functions......Page 26 2.3 Exercises......Page 32 3.1 Trigonometric Functions at the Triangle......Page 35 3.2 Extension of the Trigonometric Functions to mathbbR......Page 39 3.3 Cyclometric Functions......Page 41 3.4 Exercises......Page 43 4.1 The Notion of Complex Numbers......Page 46 4.2 The Complex Exponential Function......Page 49 4.3 Mapping Properties of Complex Functions......Page 51 4.4 Exercises......Page 53 5.1 The Notion of an Infinite Sequence......Page 55 5.2 The Completeness of the Set of Real Numbers......Page 61 5.3 Infinite Series......Page 64 5.4 Supplement: Accumulation Points of Sequences......Page 68 5.5 Exercises......Page 71 6.1 The Notion of Continuity......Page 74 6.2 Trigonometric Limits......Page 79 6.3 Zeros of Continuous Functions......Page 80 6.4 Exercises......Page 83 7.1 Motivation......Page 85 7.2 The Derivative......Page 87 7.3 Interpretations of the Derivative......Page 91 7.4 Differentiation Rules......Page 94 7.5 Numerical Differentiation......Page 100 7.6 Exercises......Page 105 8.1 Curve Sketching......Page 108 8.2 Newton's Method......Page 113 8.3 Regression Line Through the Origin......Page 118 8.4 Exercises......Page 121 9.1 Fractals......Page 125 9.2 Mandelbrot Sets......Page 132 9.3 Julia Sets......Page 133 9.4 Newton's Method in mathbbC......Page 134 9.5 L-systems......Page 136 9.6 Exercises......Page 140 10.1 Indefinite Integrals......Page 141 10.2 Integration Formulas......Page 144 10.3 Exercises......Page 148 11.1 The Riemann Integral......Page 150 11.2 Fundamental Theorems of Calculus......Page 156 11.3 Applications of the Definite Integral......Page 159 11.4 Exercises......Page 162 12.1 Taylor's Formula......Page 165 12.2 Taylor's Theorem......Page 169 12.3 Applications of Taylor's Formula......Page 170 12.4 Exercises......Page 173 13.1 Quadrature Formulas......Page 175 13.2 Accuracy and Efficiency......Page 180 13.3 Exercises......Page 182 14.1 Parametrised Curves in the Plane......Page 185 14.2 Arc Length and Curvature......Page 193 14.3 Plane Curves in Polar Coordinates......Page 200 14.4 Parametrised Space Curves......Page 202 14.5 Exercises......Page 204 15.1 Graph and Partial Mappings......Page 208 15.2 Continuity......Page 210 15.3 Partial Derivatives......Page 211 15.4 The Fréchet Derivative......Page 215 15.5 Directional Derivative and Gradient......Page 220 15.6 The Taylor Formula in Two Variables......Page 222 15.7 Local Maxima and Minima......Page 223 15.8 Exercises......Page 227 16.1 Vector Fields and the Jacobian......Page 230 16.2 Newton's Method in Two Variables......Page 232 16.3 Parametric Surfaces......Page 235 16.4 Exercises......Page 237 17.1 Double Integrals......Page 239 17.2 Applications of the Double Integral......Page 245 17.3 The Transformation Formula......Page 247 17.4 Exercises......Page 251 18.1 Simple Linear Regression......Page 253 18.2 Rudiments of the Analysis of Variance......Page 259 18.3 Multiple Linear Regression......Page 263 18.4 Model Fitting and Variable Selection......Page 265 18.5 Exercises......Page 269 19.1 Initial Value Problems......Page 272 19.2 First-Order Linear Differential Equations......Page 275 19.3 Existence and Uniqueness of the Solution......Page 280 19.4 Method of Power Series......Page 283 19.5 Qualitative Theory......Page 285 19.6 Second-Order Problems......Page 287 19.7 Exercises......Page 291 20.1 Systems of Linear Differential Equations......Page 293 20.2 Systems of Nonlinear Differential Equations......Page 304 20.3 The Pendulum Equation......Page 308 20.4 Exercises......Page 313 21.1 The Explicit Euler Method......Page 316 21.2 Stability and Stiff Problems......Page 319 21.3 Systems of Differential Equations......Page 322 21.4 Exercises......Page 323 A.1 Cartesian Coordinate Systems......Page 325 A.3 Vectors in a Cartesian Coordinate System......Page 326 A.4 The Inner Product (Dot Product)......Page 329 A.5 The Outer Product (Cross Product)......Page 330 A.6 Straight Lines in the Plane......Page 331 A.7 Planes in Space......Page 333 A.8 Straight Lines in Space......Page 334 B.1 Matrix Algebra......Page 336 B.2 Canonical Form of Matrices......Page 340 C.1 Continuity of the Inverse Function......Page 345 C.2 Limits of Sequences of Functions......Page 346 C.3 The Exponential Series......Page 349 C.4 Lipschitz Continuity and Uniform Continuity......Page 353 Supplementary Software Description......Page 356 Refs......Page 358 Index......Page 360