Introduction to the Tools of Scientific Computing

Description

Preface Preface to the Second Edition Acknowledgments Preface to the First Edition Acknowledgments Contents Chapter 1 Introduction Background Pitfalls and Limitations Programming Languages The C Language Program Packages Matlab, Maple, Mathematica The Golden Middle Course: Python Part I Background Part II Core Languages Part III Commercial Computing Environments Part IV Distributed Computing Part V Specialized Programming Frameworks Part I Background Chapter 2 Mathematical Foundations of Programming 2.1 A Simple Machine Model Machine Language and Higher-Level Programming Languages Church’s Thesis The Turing Machine Limits of Computability Repercussions in Mathematics Extended Number Ranges 2.2 Digital Computers Number Representation Natural Numbers Integer Numbers Real Numbers, Floating Point Numbers Higher-Level Programming Languages Part II Core Languages Chapter 3 Python, the Fundamentals 3.1 Python Interpreter A First Example Interactive Mode 3.2 Elementary Data Types Numbers and Arithmetic Integer Numbers Real Numbers, Floating Point Numbers Arithmetical Operations Error Propagation in float Computations The Types complex and bool Complex Numbers Boolean Operators, Comparisons 3.3 Variables and Value Assignments Basic Rule 3.4 Control Structures Loops While Loops For Loops Interlude: the range Function The break and continue Statements Conditional Statements Floating Point Precision Revisited: Machine Epsilon 3.5 Collection Types: Lists, Tuples, Dictionaries and Sets Lists list is a Reference Type Sieve of Eratosthenes Sublists Concatenation List Comprehensions Tuples Dictionaries Sets 3.6 Functions Anonymous Functions, Lambda Expressions Functions as Arguments Functions as Return Values Recursion The Quicksort Algorithm 3.7 String Interpolation and Formatting 3.8 Writing and Reading Files Writing Strings Interlude: Access Paths Reading Numbers Binary Files and pickle 3.9 Object-Oriented Programming and Classes Polynomials Inheritance Exercises Machine Epsilon Polynomial Class Linear Algebra Numerical Integration with the Trapezoidal Rule The Taylor Series The Newton Method Fermat’s Last Theorem Chapter 4 Python in Scientific Computation 4.1 NumPy NumPy Arrays Vectors and Matrices Vectors Matrices Access to Matrix Components Flatten and Reshape Standard Matrices Advanced Construction Methods The Methods diag and kron, the Poisson Matrix 4.2 Conjugate Gradient Method Quadratic Form The Method 4.3 SciPy 4.4 Linear Algebra Matrix Algorithms Linear Equations LU Decomposition Cholesky Decomposition Application: Method of Least Squares QR Decomposition QR Decomposition in Rectangular Matrices The Gaussian Normal Equation Eigenvectors and Eigenvalues Sparse Matrices Sparse Poisson Matrix 4.5 Graphics with Matplotlib 4.6 Nonlinear Equations, Optimization Single Equations Equation Systems Minimization 4.7 Numerical Integration, Ordinary Differential Equations Integration Ordinary Differential Equations ODEs Initial Value Problems, solve_ivp The Solver solve_ivp Lotka–Volterra Equations Pendulum Boundary Value Problems BVPs, solve_ivp 4.8 Partial Differential Equations Discretization Solution Program Sparse Poisson Matrix Nonhomogeneous Boundary Conditions The SciPy Program Verification 4.9 Round off: Random Numbers Monte Carlo Method for the Computation of 𝜋 Exercises Linear Equations Nonlinear Equations Newton’s Method in Rn Integration Optimization Ordinary Differential Equations, Initial Value Problems Ordinary Differential Equations, Boundary Value Problems Partial Differential Equations Chapter 5 Python in Computer Algebra 5.1 Symbolic Calculation, Numbers Number Types Symbolic Expressions Convention Value Assignments Advanced Transformations Functions 5.2 Equation Systems 5.3 Linear Algebra Matrices and Vectors Vector Space Bases Eigenvectors and Eigenvalues Matrices with 𝑛 ≥ 5 5.4 Calculus Limits Differential Calculus Integration Series Expansion 5.5 Ordinary Differential Equations 5.6 Galerkin Method The Model Equation The Variational Problem The Method The SymPy Program Exact Solution Exercises Linear Equations Eigenvalues Nonlinear Equations Differential Equations Function Spaces Chapter 6 The C Language Integrated Development Environment Command Line Compiler 6.1 Basics Numbers and Arithmetic Operations Number Representations Arithmetic Operators Output Variables and Assignments 6.2 Control Structures: Branches, Loops Conditional Execution Loops 6.3 Functions Functions as Arguments 6.4 Arrays Arrays as Function Arguments Matrices 6.5 Pointers Arrays and Pointers Pointers as Arrays 6.6 Structures Named Structures Sparse Matrices 6.7 Files, Input and Output 6.8 Conclusion Chapter 7 The C++ Language 7.1 Transfer from C 7.2 Basics The Output Operator