Foundations, Methods, and Algorithms: Computer Science Analysis

**Foundations, Methods, and Algorithms: Computer Science Analysis is a comprehensive** guide that delves into the core principles of computer science and numerical mathematics. This book covers a wide array of topics, including algorithms and their analysis, providing a solid foundation for understanding computational methods. With practical applications in mind, it incorporates tools like MATLAB and Maple to aid readers in implementing and solving problems. Whether you're a student or a professional in the field of computer science, this resource equips you with the knowledge and techniques needed to effectively analyze algorithms and tackle computational challenges. It's an essential reference for anyone seeking a deeper understanding of computer science analysis. Numbers................................................1 The Real Numbers......................................1 Order Relation and Arithmetic on R...........................5 Machine Numbers......................................8 Rounding...........................................10 Exercises...........................................11 Real-Valued Functions......................................13 Basic Notions........................................13 Some Elementary Functions...............................17 Exercises...........................................23 Trigonometry............................................27 Trigonometric Functions at the Triangle.......................27 Extension of the Trigonometric Functions to R..................31 Cyclometric Functions..................................33 Exercises...........................................35 Complex Numbers........................................39 The Notion of Complex Numbers...........................39 The Complex Exponential Function..........................42 Mapping Properties of Complex Functions.....................44 Exercises...........................................46 Sequences and Series.......................................49 The Notion of an Infinite Sequence..........................49 The Completeness of the Set of Real Numbers...................55 Infinite Series........................................58 Supplement: Accumulation Points of Sequences..................62 Exercises...........................................65 Limits and Continuity of Functions..............................69 The Notion of Continuity.................................69 Trigonometric Limits...................................74 ix Zeros of Continuous Functions.............................75 Exercises...........................................78 The Derivative of a Function..................................81 Motivation..........................................81 The Derivative........................................83 Interpretations of the Derivative............................87 Differentiation Rules....................................90 Numerical Differentiation................................96 Exercises..........................................101 Applications of the Derivative................................105 Curve Sketching......................................105 Newton’s Method.....................................110 Regression Line Through the Origin.........................115 Exercises..........................................118 Fractals and L-systems.....................................123 Fractals............................................124 Mandelbrot Sets......................................130 Julia Sets...........................................131 Newton’s Method in C..................................132 L-systems..........................................134 Exercises..........................................138 Antiderivatives..........................................139 Indefinite Integrals....................................139 Integration Formulas...................................142 Exercises..........................................146 Definite Integrals.........................................149 The Riemann Integral..................................149 Fundamental Theorems of Calculus.........................155 Applications of the Definite Integral.........................158 Exercises..........................................161 Taylor Series...........................................165 Taylor’s Formula.....................................165 Taylor’s Theorem.....................................169 Applications of Taylor’s Formula..........................170 Exercises..........................................173 Numerical Integration.....................................175 Quadrature Formulas...................................175 Accuracy and Efficiency................................180 Exercises..........................................182 Curves...............................................185 Parametrised Curves in the Plane...........................185 Arc Length and Curvature...............................193 Plane Curves in Polar Coordinates..........................200 Parametrised Space Curves...............................202 Exercises..........................................204 Scalar-Valued Functions of Two Variables........................209 Graph and Partial Mappings..............................209 Continuity..........................................211 Partial Derivatives....................................212 The Fréchet Derivative.................................216 Directional Derivative and Gradient.........................221 The Taylor Formula in Two Variables.......................223 Local Maxima and Minima...............................224 Exercises..........................................228 Vector-Valued Functions of Two Variables.......................231 Vector Fields and the Jacobian............................231 Newton’s Method in Two Variables.........................233 Parametric Surfaces...................................236 Exercises..........................................238 Integration of Functions of Two Variables........................241 Double Integrals......................................241 Applications of the Double Integral.........................247 The Transformation Formula.............................249 Exercises..........................................253 Linear Regression........................................255 Simple Linear Regression................................255 Rudiments of the Analysis of Variance.......................261 Multiple Linear Regression...............................265 Model Fitting and Variable Selection........................267 Exercises..........................................271 Differential Equations.....................................275 Initial Value Problems..................................275 First-Order Linear Differential Equations.....................278 Existence and Uniqueness of the Solution.....................283 Method of Power Series.................................286 Qualitative Theory....................................288 Second-Order Problems.................................290 Exercises..........................................294 Systems of Differential Equations..............................297 Systems of Linear Differential Equations.....................297 Systems of Nonlinear Differential Equations...................308 The Pendulum Equation.................................312 Exercises..........................................317 Numerical Solution of Differential Equations......................321 The Explicit Euler Method...............................321 Stability and Stiff Problems..............................324 Systems of Differential Equations..........................327 Exercises..........................................328 Appendix A: Vector Algebra.................................331 Appendix B: Matrices.....................................343 Appendix C: Further Results on Continuity.......................353 Appendix D: Description of the Supplementary Software..............365 References.............................................367 Index................................................369