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.
Author(s): Educohack Press
Publisher: Educohack Press
Year: 2023
Language: English
Pages: 1064
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
